TEXT-BOOKS   OF    PHYSICAL 
CHEMISTRY 

EDITED  BY  SIR  WILLIAM  RAMSAY,  K.C.B.,  F.R.S. 

A   SYSTEM   OF    PHYSICAL 
CHEMISTRY 


TEXT-BOOKS  OF  PHYSICAL  CHEMISTRY 

EDITED  BY  SIR  WILLIAM  RAMSAY,  K.C.B.,  D.Sc.,  F.R.S. 

Crown  8vo. 

STOICHIOMETRY.  By  SYDNEY  YOUNG,  D.Sc.,  F.R.S.,  Professor 

of  Chemistry  in  the  University  of  Dublin  ;  together  with  Ax  INTRODUCTION 

TO  THE  STUDY  OF  PHYSICAL  CHEMISTRY  by  Sir  WILLIAM   RAMSAY, 

K.C.B.,  D.Sc.,  F.R.S.    6s.  Gd.  net. 
CHEMICAL    STATICS     AND     DYNAMICS,    including    the 

Theories  of  Chemical  Change,    Catalysis   and   Explosions.     By  T.   W. 

MELLOR,  D.Sc.,  B.Sc.    6s.  6d.  net. 
THE  PHASE  RULE  AND  ITS  APPLICATIONS.     By  ALEX. 

FINDLAY,  M.A.,  Ph.D.,  D.Sc.     With  134  Figures  in  the  Text.     FOURTH 

EDITION.    6^.  net. 
SPECTROSCOPY.      By   E.   C.   C.   BALY,   F.I.C ,   Professor  of 

Chemistry  in  the  University  of  Liverpool.    With  180  Illustrations.     NEW 

EDITION,     los.  6d,  net. 

THERMOCHEMISTRY.  By  JULIUS  THOMSEN,  Emeritus  Pro- 
fessor of  Chemistry  in  the  University  of  Copenhagen.  Translated  by 
KATHARINE  A.  BURKE,  B.Sc.  (Lond.),  Department  of  Chemistry, 
University  College,  London,  js.  6d.  net. 

STEREOCHEMISTRY.     By    ALFRED    VV.    STEWART,  D.Sc., 

Carnegie  Research  Fellow,  Lecturer  on  Chemistry  at  Queen's  University, 
Belfast.    With  87  Illustrations,    gs.  net. 

ELECTRO-CHEMISTRY.  PART  I.— GENERAL  THEORY.  By 
R.  A.  LEHFELDT,  D.Sc.,  Professor  of  Natural  Philosophy  and  Physics  at 
the  Transvaal  University  College,  Johannesburg.  Including  a  Chapter  on 
the  Relation  of  Chemical  Constitution  to  Conductivity.  By  T.  S.  MOORE, 
B.A.,  B.Sc.,  Lecturer  in  the  University  of  Birmingham.  NEW  IMPRESSION. 
4_y.  6d.  net. 

ELECTRO-CHEMISTRY.    PART  II.    By  E.  B.  R.  PRIDEAUX, 

D.Sc.,  of  Battersea  Polytechnic,  S.W.  [In  preparation. 

THE  THEORY  OF  VALENCY.  By  J.  NEWTON  FRIEND, 
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METALLOGRAPHY.  By  CECIL  H.  DESCH,  D.Sc.  (Lond.), 
Ph.D.  (Wiirz.),  Graham  Young  Lecturer  in  Metallurgical  Chemistry  in 
the  University  of  Glasgow.  With  14  Plates  and  108  Diagrams  in  the 
text.  SECOND  EDITION,  js.  6d.  net. 

THE  RELATIONS  BETWEEN  CHEMICAL  CONSTI- 
TUTION AND  SOME  PHYSICAL  PROPERTIES.  By  SAMUEL 
SMILES,  D.Sc.,  Fellow  of  University  College,  and  Assistant  Professor  of 
Organic  Chemistry  at  University  College,  London  University.  123.  net. 

PHOTOCHEMISTRY.  By  S.  E.  SHEPPARD,  D.Sc.,  formerly 
1851  Exhibition  Research  Scholar  of  University  College,  London.  With 
27  Illustrations,  etc.  los.  6d.  net. 

A  SYSTEM  OF  PHYSICAL  CHEMISTRY.  By  W.  C.  McC. 
LEWIS,  M.A.,  D.Sc.,  Brunner  Professor  of  Physical  Chemistry  in  the 
University  of  Liverpool.  With  Diagrams.  2  vols.  gs.  net  per  volume. 
(Sold  separately.) 

PRACTICAL   SPECTROGRAPHIC   ANALYSIS.      By  J.   H. 

POLLOK,  D.Sc.  \_Inpreparation. 

CRYSTALLOGRAPHY.  By  T.  V.  BARKER,  B.Sc.,  M.A.,  Fellow 
of  Brasenose  College,  Oxford.  [In  preparation. 

LONGMANS,    GREEN   AND   CO. 

LONDON,    NEW  YORK,    BOMBAY,    CALCUTTA,    AND   MADRAS 


A    SYSTEM    OF 

PHYSICAL  CHEMISTRY 


IN   TWO    VOLUMES 


BY 

WILLIAM  C.  MCC.  LEWIS,  M.A.  (R.U.I.),  D.Sc.(Liv.) 

BRUNNER   PROFESSOR   OF  PHYSICAL   CHEMISTRY   IN   THE   UNIVERSITY  OF 

LIVERPOOL  ;    FORMERLY  LECTURER    IN   PHYSICAL  CHEMISTRY 

UNIVERSITY  COLLEGE,    LONDON 


WITH  98   DIAGRAMS  IN  THE    TEXT 


VOLUME    II 


LONGMANS,    GREEN    AND    CO. 
39    PATERNOSTER    ROW,    LONDON 

FOURTH   AVENUE  &   3oTH   STREET,   NEW  YORK 
BOMBAY,    CALCUTTA,    AND    MADRAS 


All  rights  renewed 


:  ; 


TABLE  OF  CONTENTS  OF  VOL.  II 
PART    II 

CONSIDERATIONS  BASED  UPON  THERMODYNAMICS 
CHAPTER    I 

PAGE 

Introductory — Elementary  consideration  of  the  principles  of  thermo- 
dynamics    i 

CHAPTER   II 

Introductory — Further  consideration  of  thermodynamic  principles    .       49 

CHAPTER   III 

Continuity  of  the  liquid  and  gaseous  states  from  the  thermodynamic 

standpoint 75 

CHAPTER   IV 

Thermodynamic  criteria  of  chemical  equilibrium  in  general    .      .      .     1 1 1 

CHAPTER   V 

Chemical  equilibrium  in  homogeneous  systems  from  the  thermo- 
dynamic standpoint — Gaseous  systems — Deduction  of  the  law  of 
mass  action — The  van  't  Hoff  isotherm — Principle  of  "mobile 
equilibrium "  (Le  Chatelier  and  Braun) — Variation  of  the 
equilibrium  constant  with  temperature 132 

349283 


yi  TABLE    OF  CONTENTS 

CHAPTER   VI 

FACE 

Chemical  equilibrium  in  homogeneous  systems — Dilute  solutions — 
Applicability  of  the  gas  laws — Thermodynamic  relations  between 
osmotic  pressure  and«the  lowering  of  vapour  pressure,  the  rise  of 
boiling  point,  the  lowering  of  freezing  point  of  the  solvent,  and 
change  in  the  solubility  of  the  solvent  in  another  liquid— Mole- 
cular weight  of  dissolved  substances — Law  of  mass  action — 
Change  of  equilibrium  constant  with  temperature  and  pressure  .  150 


CHAPTER   VII 

Chemical    equilibrium   in   homogeneous   systems — Dilute    solutions 

(continued} — Outlines  of  the  electrochemistry  of  dilute  solutions  .      1 74 


CHAPTER   VIII 

Chemical    equilibrium     in     homogeneous     systems  —  Concentrated 

solutions .  216 


CHAPTER   IX 

Chemical  equilibrium  in  heterogeneous  systems  in  the  absence  of 
electrical,  capillary,  or  gravitational  effects — The  Phase  Rule  and 
some  of  its  applications — The  theory  of  allotropy  ....  238 

CHAPTER   X 

Chemical  equilibrium  in  heterogeneous  systems  from  the  thermo- 
dynamic  standpoint  when  capillary  or  electrical  effects  are  of  im- 
portance— Adsorption — Donnan's  theory  of  membrane  equilibria  303 

CHAPTER   XI 

Systems  not  in  equilibrium  from  the  thermodynamic  standpoint — 
Affinity  and  its  measurement  by  means  of  vapour  pressure, 
solubility,  and  electromotive  force — Oxidation  and  reduction 
processes — Change  of  affinity  with  temperature  .  .  ,  .  .  320 

CHAPTER   XII 

Systems  not  in  equilibrium  (continued) — Relation  between  the  affinity 
and  the  heat  of  a  reaction— Nernst's  heajtjjheorem  and  some  of 
its  applications 357 


TABLE   OF  CONTENTS 


PART    III 

CONSIDERATIONS   BASED   UPON   THERMODYNAMICS 
AND  STATISTICAL  MECHANICS 

CHAPTER    I 

PAGE 

Behaviour  of  systems  (in  equilibrium  and  not  in  equilibrium),  exposed 
to  radiation — Photochemistry — Thermodynamic  treatment  of 
photochemical  reactions 397 

CHAPTER   II 

Applications  of  the  Unitary  Theory  of  Energy  (Energy-quanta)  to 

physical  and  chemical  problems 463 


SUBJECT  INDEX 543 

AUTHOR  INDEX     549 


A   SYSTEM   OF 

PHYSICAL    CHEMISTRY 

PART   II 

CONSIDERATIONS   BASED   UPON   THERMO- 
DYNAMICS 

CHAPTER    I 

ELEMENTARY  CONSIDERATION  OF  THE  PRINCIPLES  OF 
THERMODYNAMICS. 

THE  science  of  thermodynamics  deals  with  the  laws  and 
relationships  which  govern  the  quantitative  transformations  of 
energy  of  one  sort  into  energy  of  some  other  sort  during 
physical  or  chemical  changes  in  a  system.  The  name  suggests 
the  mutual  relation  of  heat  and  motion,  but  the  principles 
involved  have  a  much  larger  scope,  and  may  be  applied  under 
given  conditions  to  all  forms  of  energy.  The  characteristic 
feature  of  thermodynamics  is  that  it  permits  us  to  deal  with 
energy  changes  involved  in  a  physical  change  of  state,  or  in  a 
chemical  reaction  without  in  any  way  requiring  information 
regarding  the  molecular  mechanism  of  the  process  under  investi- 
gation. The  conclusions  which  we  shall  arrive  at  on  the  basis 
of  thermodynamics  are  thus  independent  of  any  molecular 
hypothesis  we  may  have  formed  in  respect  of  the  process. 
This,  it  will  be  seen,  is  in  many  ways  a  very  great  advantage. 
It  means  that  the  conclusions  of  thermodynamics  are  quite 
general,  and  will  remain  true  even  if  our  views  regarding  the 
T.P.C. — ii.  B 


2  A   SYSTEM  OF '.PHYSICAL    CHEMISTRY 

actual  mechanism  of  the  process1  considered  from  a  molecular 
standpoint  were  to  undergo  radical  change.  Of  course  it  will 
be  seen  as  the  converse  of  this  that  thermodynamical  reasoning 
and  conclusions,  althoifgh  true  and  general,  do  not  tell  us 
anything  of  the  mechanism  involved  in  a  given  process.  This 
is  naturally  a  considerable  drawback,  for  advances  in  theoretical 
treatment  seem  to  be  most  easily1  made  along  mechanical  lines 
of  thought,  i.e.  with  the  aid  of  molecular  hypotheses.  At 
the  present  time,  therefore,  the  line  of  attack  upon  any  prob- 
lem which  promises  to  be  most  successful  is,  in  general,  that 
in  which  we  make  simultaneous  use  of  both  generalised 
principles  (thermodynamics)  and  the  specialised  principles 
(such  as  the  Kinetic  Theory).  In  Part  I.  of  this  book  the 
problems  of  equilibrium  in  physical  and  chemical  systems,  and 
the  behaviour  of  systems  not  in  equilibrium,  have  been  studied 
from  a  molecular  standpoint.  In  the  present  part  we  shall 
study  the  same  problems  from  the  standpoint  of  thermo- 
dynamics, bringing  out  as  far  as  possible  the  general  relation 
of  thermodynamical  conclusions  to  the  conclusions  drawn  with 
the  help  of  the  Kinetic  Theory.  It  is,  of  course,  necessary 
first  of  all  to  obtain  some  information  respecting  the  general 
ideas  underlying  the  science  of  Energetics  or  Thermodynamics, 
and  this  matter  forms  the  subject  of  the  present  chapter. 

The  classical  theory  of  thermodynamics  may  be  said  to 
rest  upon  two  main  laws,  or  fundamental  principles,  known  as 
the  First  and  Second  Laws  respectively.  Recently  another 
principle  has  been  brought  forward  by  the  noted  German 
scientist,  Professor  Nernst,  of  Berlin,  which  seems  likely  to 
ultimately  take  its  place  as  a  third  law.  This  principle  and 
its  applications  are  discussed  at  the  conclusion  of  our  study  of 
affinity  (Part  II.,  Chap.  XII.).  For  the  present  we  shall 
confine  our  attention  to  the  First  and  Second  Laws. 

THE  FIRST  LAW  OF  THERMODYNAMICS. 

This  law  is  simply  a  statement  of  the  principle  known  as 
the  Conservation  of  Energy,  according  to  which  energy  can 
be  changed  from  one  form  to  another,  but  can  never  be 


CONSERVATION  OF  ENERGY  3 

destroyed.     Before  discussing  this  let  us  consider  briefly  what 
we  mean  by  the  term  "  energy." 

When  a  body  or  system  can  do  work  against  a  force  it  is  said 
to  possess  energy.  How  are  we  to  obtain  a  measure  of  this  energy  ? 
This  is  effected  by  regarding  the  decrease  in  energy  as  equal 
to  the  work  done.     When  the  energy  arises  from  motion — say 
of  the  molecules  of  a  system,  or  of  the  system  as  a  whole  in 
space — it  is  called  Kinetic  Energy.     When  it  arises  from  the 
relative  position  of  bodies  it  is  called  Potential  Energy.     From 
the  standpoint  of  thermodynamics,  however,  we  do  not  dis- 
tinguish the  origin  of  energy  whether  it  be  kinetic  or  potential 
(as  this  would  really  involve  molecular  hypotheses) ;  instead, 
we  deal  simply  with  the  energy  possessed  by  a  body,  or  more 
frequently  with  the  energy  gained  or  lost  by  a  system  when 
the  system  undergoes  a  given  change,  without  specifying  more 
closely  whether  the  energy  thus  lost  or  gained  is  kinetic  or 
potential  in  nature.     It  is  important  to  notice  that  although 
we  can  mentally  conceive  of  the  idea  of  a  given  substance  or 
system  possessing  energy,  we  are  unable  to  give  this  a  numerical 
value.     What  we  can  do,  however,  is  to  ascribe  a  numerical 
value  to  the  change  in  energy  involved  in  a  given  process  j  and 
it  is  with  such  changes  of  energy^  i.e.  either   gain    or   loss  of 
energy,  that  thermodynamics  deals.     A  simple  illustration  of 
a  mechanical  nature  will  make  the  distinction  clear.     Suppose 
a  body  having  the  mass  of  i  gram  falls  a  distance  of  i  meter 
towards  the  earth  under  the  action.of  gravity.    The  loss  of  poten- 
tial energy  sustained  by  the  body  in  thus  altering  its  position 
is  i  gram-meter,  this  being  the  work  done  in  the  change  con- 
sidered.    We  do  not  know  the  absolute  value  of  the  potential 
energy  of  the  body  in  its  initial  position,  but  we  do  know  that 
in  its  final  position  it  has  lost  an  amount,  i  gram-meter.     Now, 
the  study  of  mechanics  has  made  us  familiar  with  the  idea  that 
the  term  "  work,"  or  "  energy,"  is  a  composite  one,  being  always 
expressible  as  the  product  of  two  terms,  one  term  being  known 
as  the  "  intensity  factor,"  the  other  as  the  "  capacity  factor." 
The  simplest  conception  of  work  or  energy  is  to  regard  it  as 
the  product  of  force  into  distance.     The  force  is  the  "  intensity 
factor,"  the  distance  the  "  capacity  factor."     A  force  has  the 


4  A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

dimensions1   ^  ,  and  a  distance   the  dimension  L,  so  that 

energy  or  work  must  ha^ve  the  dimensions  -^~'     Even  when 

the  actual  energy  is  not  simply  force  X  distance  it  must  still 
have  the  dimensions  above  ascribed  to  it.  Thus  let  us  con- 
sider another  sort  of  simple  process  in  which  work  is  done 
and  the  equivalent  energy  expended,  namely,  the  isothermal 
expansion  of  a  gas  against  the  atmosphere  or  other  pressure. 
Suppose  the  initial  volume  of  the  gas  was  v  c.c.,  and  the 
increase  in  volume  is  a  very  small  one  dv.  Suppose  p  is  the 
atmospheric  pressure  (in  dynes  per  square  centimeter).  Let 
us  suppose  the  gas  is  in  a  cylinder  fitted  with 
a  weightless,  frictionless,  movable  piston  as  in 
Fig.  38.  The  surface  area  of  the  piston  may 
be  A  cm2.  In  the  equilibrium  state  the  pressure 
above  and  below  the  piston  must  be  identical 


A 

Gas 


dynes\ 


.    /        2    ).      Under    these    conditions    the 
cm2.  / 

piston  would  remain  motionless  for  an  infinite 
time.  Suppose,  however,  that  the  gas  pressure, 
i.e.  the  pressure  exerted  upon  the  lower  side  of  the  piston, 
is  something  greater  than  p.  The  gas  will  tend  to  push  the 
piston  outwards.  Suppose  it  does  so  through  an  infinitely 
small  distance  dx.  The  work  done  =  force  X  distance 
=  (/  X  A)  X  dx,  but  MX  —  dv;  .'.  work  =  pdv.  Now, 
in  performing  this  expansion  against  the  external  pressure  /, 
the  gas  has  done  work  upon  the  surrounding  atmosphere, 
or,  in  other  words,  it  has  caused  energy  to  be  expended.  We 
shall  speak  of  the  source  of  this  energy  in  a  moment.  The 
expression  for  the  work  done  is  simply  the  pressure  into  the 
increase  in  volume,  namely,/^.  We  see  that  this  term  actually 
represents  work,  for  it  has  the  correct  dimensions,  namely, 

mass   x    velocity 

1  Force  is   mass    X    acceleration    =    -        — : —     — —  and   velocity 

time 

=  —    —    Denoting  the  dimensions  of  mass  by  M,  length  by  L,  and  time 
by  T,  we  can  write  the  dimensions  of  force  as  being  - — =^ — • 


WORK  OF  EXPANSION  (OF  A    GAS}  5 

ML2 

-rpa~>  and  ma7  be  ultimately  expressed  as  force  X  distance. 

The  source  of  the  energy  which  has  been  used  in  the  work  of 
expansion  if  the  temperature  has  been  maintained  constant,  is 
the  heat  which  has  been  drawn  from  the  surroundings.  This 
is  an  "  isothermal  expansion."  If  we  had  exclosed  the  gas  in 
a  "  heat  tight "  case  and  allowed  it  to  do  work  we  would  have 
found  that  its  temperature  would  have  fallen  (though  only  to 
an  infinitely  small  extent,  if  the  expansion  was  infinitely  small). 
In  the  latter  case  the  work  is  done  at  the  expense  of  the  heat 
energy  of  the  gas  itself.  By  this  heat  energy  we  mean  the 
kinetic  energy  of  the  molecules,  for  heat,  properly  speaking, 
does  not  reside  in  a  system,  being  transferable  from  one  system 
to  another,  but  always  undergoing  transformation  into  some- 
thing else  (i.e.,  say,  into  kinetic  energy  of  the  molecules)  after 
its  addition  to  a  system.  It  will  be  observed  that  in  order  to 
keep  the  temperature  of  the  gas  constant  whilst  expansion 
proceeds  we  have  assumed  that  heat  is  given  to  the  gas  from 
its  surroundings  in  order  to  compensate  for  the  cooling  effect 
of  the  expansion  itself.  This  really  involves  the  principle  of 
the  First  Law,  for  what  we  have  actually  done  is  to  transform 
heat  energy  into  mechanical  energy ;  the  gas  simply  acting  as 
the  medium. 

Besides  heat  and  mechanical  energy  we  are  acquainted 
with  many  other  forms  of  energy,  i.e.  electrical  energy,  mag- 
netic energy,  radiant  energy,  surface  energy  (i.e.  the  product 
of  surface  tension  of  a  liquid  into  the  area  of  the  surface), 
and  chemical  energy.  All  these  must  ultimately  be  expres- 
sible by  the  expression  -7™--  Now,  the  First  Law  deals  with 

the  transformation  of  any  kind  of  energy  into  any  other  kind, 
and  according  to  the  law,  when  a  certain  amount  of  one  kind 
disappears  an  exactly  equivalent  quantity  of  some  other 
kind  must  appear.  When  a  bullet  strikes  a  target  there  is  heat 
produced,  the  amount  of  heat  being  equivalent  to  the  kinetic 
energy  which  the  bullet  possessed  just  before  striking.  If  we 
lose  100  ergs  of  mechanical  energy  in  some  process  we  must 
gain  100  ergs  in  some  other  form — heat,  for  example ;  for  energy 


6  A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

can  neither  be  created  nor  destroyed  as  a  whole.  This  is  why 
perpetual  motion  is  impossible ;  it  is  more  accurate  to  say  that 
in  the  experience  of  mankind,  no  instance  of  perpetual  motion 
has  been  recorded,  and  we  have  therefore  concluded  that  such 
is  impossible.  This  is  stated  by  the  First  Law.  The  First  Law 
is  a  law  of  experience,  just  as  the  Second  Law  is,  as  we  shall 
see  later.  On  the  experimental  side  the  First  Law  has  received 
quantitative"  confirmation  by  the  extremely  accurate  determi- 
nations of  the  so-called  "  mechanical  equivalent  of  heat  " 
carried  out  in  the  first  instance  by  Joule,  and  later  by  Rowlands 
and  others.  Joule's  determination  of  the  equivalent  consisted 
in  placing  some  water  in  an  isolated  vessel  and  then  rapidly 
rotating  a  paddle  in  the  water.  He  observed  that  the  tem- 
perature of  the  water  rose — -just  as  Rumford,  many  years 
previously,  had  observed  that  in  the  boring  of  a  cannon  the 
friction  caused  the  metal  to  become  extremely  hot.  With 
Joule's  arrangement  it  was  possible  to  obtain  a  quantitative 
connection  between  the  mechanical  work  done  in  stirring,  and 
the  heat  added  to  the  water,  as  evidenced  by  the  rise  in  tem- 
perature. The  paddle  was  worked  by  means  of  a  falling  weight, 
the  working  agency  being  therefore  gravitation.  Frictional 
effects  were  reduced  as  far  as  possible.  Knowing  the  weight 
of  the  falling  body,  and  the  distance  through  which  it  fell,  the 
amount  of  work  done  by  gravitation  can  be  directly  obtained. 
If  now  the  rise  in  temperature  of  the  water  is  measured, 
the  heat  capacity  of  the  water  being  known,  the  amount  of 
heat  in  calories  generated  by  the  motion  of  the  paddle  is  also 
obtained.  If  we  neglect  or  allow  for  friction  and  other  dis- 
turbing effects  we  can  find  how  many  mechanical  work  units, 
i.e.  ergs,  are  equivalent  to  i  heat  unit,  i.e.  one  calorie.  Joule 
found  that  i  calorie  =  4-2  X  io7  ergs  (approx.).  Rowland's 
more  accurate  determinations  agreed  closely  with  Joule's 
value.  In  the  cases  where  work  is  done  by  the  expansion  of 
a  system  (against  pressure),  we  have  assumed  that  the  tem- 
perature can  be  kept  constant  by  adding  the  requisite  amount 
of  heat  which  is  transformed  into  mechanical  work  (pdv)  via 
the  system  considered.  That  is  to  say,  we  assume  the  validity 
of  the  First  Law  regarding  the  mutual  transformation  of  heat 


INTERNAL   ENERGY  7 

into  mechanical  energy,  and  vice  versa*  To  give  precision  to 
our  statement  of  the  First  Law  we  want  to  be  able  to  express 
it  algebraically.  For  this  purpose  let  us  think  of  some  system 
(solid,  liquid,  gaseous,  homogeneous  or  heterogeneous)  which 
undergoes  some  physical  or  chemical  change.  In  general 
there  will  be  associated  with  this  matter  change  the  following 
energy  changes. 

(1)  A  certain  amount  of  heat  may  be  absorbed  or  evolved 
by  the  system. 

(2)  A  certain  amount  of  external  work  is  either  performed 
by  the  system  or  upon  it  (by  the  surroundings). 

(3)  The  internal  energy  of  the  system  may  have  increased 
or    decreased.       (This    naturally    possesses    a     complicated 
mechanism.     Often  it  can  be  regarded  as  due  to  changes  in  the 
relative  position,  rate,  and  kind  of  motion  of  the  molecules. 
We  are,  however,  not  concerned  here  with  the  mechanism  of 
changes   going   on   in  the  interior  of  a  system.     We   simply 
consider  the  fact  itself  that  the  internal  energy,  or,  as  it  is  often 
called,  the  total  energy,  has  altered  in  a  given  process.)     No 
matter  what  the  process  itself  may  be,  the  First  Law  tells  us 
that   decrease   in   internal   energy   (supposing  there  to  be  a 
decrease,  which  decrease  we  denote  by  +  U)  must  be  equal  to 
the  external  work  done,  namely,  A  (say  due  to  an  increase  in 
volume  against  a  pressure)  plus  the  amount  of  heat  evolved 
or  lost  (call  the  heat  absorption  +  Q,  then  —  Q  represents  the 
heat  evolved).     Algebraically  this  statement  takes  the  form  — 

U  =  A  +  (-Q)orU  =  A-Q      .     .     .     (i) 

Consider  a  system  in  which  a  chemical  change  occurs  with- 
out alteration  in  volume,  and  without  doing  any  form  of  external 
work.  Suppose  the  internal  energy  decreases  by  the  amount 
U.  This  energy  leaves  the  system  in  the  form  of  heat  evolved, 
which  is  denoted  by  the  term  —  Q,,  the  suffix  v  indicating 
that  the  volume  has  been  kept  constant.  Then 


Even  when   the   reaction   cannot   be  carried   out  without  a 
change   in   volume   (say    an    increase   in   volume)    the   heat 


8  A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

equivalent  of  the  work  done  by  the  system  in  expanding, 
against  the  atmosphere,  say,  can  be  calculated,  and  this  amount 
is  added  to  the  observed  heat  evolved  to  give  the  quantity  —  Qv. 
The  symbol  Qw  must  not  be  confused  with  Q.  As  a  matter 
of  fact,  Q  is  in  many  cases  (though  not  always)  a  much  smaller 
quantity  than  Qt.  Q  is  often  called  a  "  latent  heat."  If  a 
process  occurs  and  does  work  A  of  any  kind  (not  necessarily 
mechanical),  and  at  the  same  time  the  internal  energy  of  the 
system  diminishes  by  the  amount  U,  then  the  term  Q  simply 
stands  for  the  difference  of  the  two  terms  A  and  U.  Q  is  not  a 
measure  of  U  at  all.  In  some  cases  (A  —  U)  may  be  almost 
zero,  i.e.  A  and  U  may  be  nearly  equal,  and  therefore  Q  may 
be  nearly  zero,  and  this  may  be  so  when  both  A  and  U 
(or  —  Qv)  possess  large  numerical  values.  In  other  cases, 
however,  U  (or  —  Qv)  may  be  very  small,  in  which  case  the 
work  done  (A),  which  may  be  great,  is  at  the  expense  of 
the  heat  taken  in,  viz.  Q.  This  is  very  nearly  realised  in  the 
work  done  by  an  expanding  gas.  Naturally  these  terms  must 
be  expressed  in  the  same  units,  say  calories.  Note  also,  that 
in  the  above  nomenclature  +  U  represents  a  decrease  in  the 
internal  energy  and  —  U  therefore  represents  an  increase  in 
the  internal  energy  of  the  system.  (The  sign  here  has  signi- 
ficance with  respect,  not  to  the  system  itself,  but  to  the  sur- 
roundings. A  gain  in  energy  to  the  surroundings  must  mean 
loss  in  energy  to  the  system.)  The  term  +  A  represents 
external  work  done  by  the  system  on  the  surroundings,  —  A 
represents  external  work  done  upon  the  system  by  the  sur- 
roundings. 

+  Q  represents  heat  added  to  or  absorbed  by  the  system. 
—  Q  represents  heat  taken  from  or  evolved  by  the  system. 

THE  CONCEPT  OF  Maximum  WORK. 

In  returning  to  the  question  of  the  work  done  in  the 
expansion  of  a  gas  it  will  be  observed  that  we  considered  an 
increase  in  volume  so  small  that  the  pressure  of  the  gas 
had .  remained  constant.  Suppose  now  that  we  consider  a 
finite  change  in  volume  from  v±  to  v2.  Since  the  system  is 


CONCEPT  OF  MAXIMUM    WORK  9 

a  gas  we  know  that  on  increasing  its  volume  its  own  pressure 
will  decrease.  The  work  which  can  be  got  out  of  the  gas  will 
depend  upon  the  opposing  pressure,  and  it  is  clear  that  it 
depends  upon  the  magnitude  of  this  opposing  pressure  whether 
the  gas  can  expand  up  to  v2  or  not.  If  the  opposing  pressure 
is  always  less  than  that  possessed  by  the  gas,  even  at  the 
large  volume  vZt  the  piston  on  being  released  will  move  rapidly 
from  v±  to  z>2>  tne  work  done  being  p(v%  —  v±).  But  this  work 
is  not  in  any  sense  a  definite  amount  characteristic  of  the 
pressure  of  the  gas,  for  p  is  the  external  opposing  pressure.  It 
is  evident  that  the  work  done  by  the  gas  can  vary  (even  when 
expanding  between  the  same  volume  limits)^  depending  on  the 
values  of  the  opposing  pressures.  If  the  opposing  pressure  is 
zero  the  work  will  be  zero.  Is  there  an  upper  limit  to  this 
work  term  ?  To  get  at  this  we  have  to  consider  the  work 
done  from  the  standpoint  of  the  pressure  possessed  by  the  gas 
itself.  Let  us  first  of  all  ask  the  question,  What  are  the  con- 
ditions under  which  a  gas  must  be  placed  in  order  that  a  very 
small  expansion  dv  may  be  accompanied  by  a  maximum 
output  of  work  ?  The  necessary  condition  which  must  be  fulfilled 
so  that  the  gas  can  do  a  maximum  amount  of  work  in  expanding 
by  dv  is  that  the  external  pressure  should  be  less  than  the  pressure 
(call  this  now  p)  possessed  by  the  gas  by  an  infinitely  small 
amount  dp.  If  we  imagine  the  outer  pressure  (that  of  the 
atmosphere)  to  be  (/  —  dp],  then  the  pressure  of  the  gas  p  will 
be/tfrfable  to  overcome  this  outer  pressure.  The  work  done 
is  (p  —  dp)  dv,  or,  what  is  the  same  thing,  pdvt  for  the  product 
dp.dv  is  only  of  the  second  order  of  magnitude,  being  the 
product  of  two  infinitely  small  quantities.  If  the  external 
pressure  had  been  exactly  equivalent  to  p  the  piston  would 
have  remained  motionless.  If  the  external  pressure  had  been 
p  -f-  dp  the  piston  would  have  moved  inwards  infinitely  slowly, 
the  work  done  upon  the  gas  being  —  (p  -\-  dp]dv  or  —pdv, 
the  negative  sign  denoting  a  decrease  in  the  volume  of  the  gas. 
It  will  be  seen  that  under  the  above  pressure  conditions  the 
process  is  a  reversible  one  in  the  sense  that  for  the  same  volume 
change  (in  one  case  an  increase,  in  the  other  a  decrease)  the 
work  done  is  represented  by  the  same  mimerical  magnitude 


io          A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

±  pdv.  When  the  expansion  takes  place  in  this  way  it  is 
accompanied  by  maximum  work,  so  that  the  production  of 
maximum  work  is  the  characteristic  of  a  process  which  is  being 
carried  out  in  a  reversible  manner.  We  shall  speak  at  greater 
length  of  reversible  processes  later  on.  It  may  here  be  noted 
that  since  the  external  pressure  is  so  arranged  that  the  pres- 
sure possessed  by  the  gas  is  just  able  to  overcome  it,  the  ex- 
pansion through  dv  must  take  place  infinitely  slowly.  Hence 
the  gas  does  not  get  up  momentum  which  on  stopping  would 
be  transformed  into  heat  of  indeterminate  amount.  No 
friction  effects  take  place  either  on  the  outgoing  or  return 
journey.  The  process  can,  therefore,  be  repeated  an  infinite 
number  of  times  without  any  permanent  drain  on  the  energy 
resources  of  the  system  and  surroundings.  As  already  men- 
tioned the  source  of  the  energy  which  enables  the  gas  to 
expand  and  do  work  and  yet  keep  the  temperature  constant  is 
then  "  the  heat  content  "  of  the  surroundings.  In  a  reversible 
process  the  heat  taken  in  from  the  surroundings  during  the 
volume  expansion  is  exactly  the  same  in  amount  as  the  heat 
given  back  to  the  surroundings  when  the  gas  contracts  iso- 
thermally  and  infinitely  slowly  to  its  original  position.  If 
there  had  been  a  net  loss  or  gain  of  heat,  say,  as  a  result  of 
the  total  operation,  the  process  is  called  an  irreversible  one,  for 
on  completing  the  process  isothermally  there  would  be  a  per- 
manent change  in  the  "  heat  content "  of  the  system  and 
surroundings.  A  reversible  process  is  essentially  one  in  which 
the  change  takes  place  in  a  known  and  definite  manner,  there 
being  no  "  accidental "  energy  transformations  taking  place, 
dependent  on  the /<?//?  followed,  i.e.  dependent  on  the  physical 
structure,  shape,  or  properties  of  the  system.  Of  course  a 
reversible  process  is  only  a  limiting  case.  It  cannot  be 
realised  in  practice  since  a  "  frictionless  piston  "  an  "  infinitely 
slow  process  "  with  a  "  pressure  difference  infinitely  small"  on 
the  sides  of  the  piston  are  only  mentally  realisable.  The 
significance  of  reversible  processes  and  maximum  work  pro- 
duction will  be  clear  when  we  come  later  to  study  the  Second 
Law  of  Thermodynamics. 

We  have  been  considering  how  the  maximum  work  can  be 


THERMODYNAMIC  REVERSIBILITY  n 

done  by  a  gas  expanding  through  an  infinitely  small  volume 
dv.  We  have  now  to  consider  the  maximum  work  produced 
when  a  gas  undergoes  a  measureable  volume  increase  from 
v-i  to  vz.  We  know  that  the  pressure  of  a  gas  falls  as  the 
volume  increases.  We  cannot,  therefore,  use  the  expression 
p(v%  —  e'j)  as  the  expression  for  the  maximum  work.  In  fact, 
we  cannot  think  of  the  external  pressure  on  the  piston  remain- 
ing constant.  It  must  also  continuously  decrease  in  the  same 
proportion  as  that  of  the  gas  pressure  itself,  being  at  any  stage 
only  less  than  that  of  the  gas  itself  by  the  infinitely  small 
amount  dp.  To  find  out  what  the  maximum  work  is  we  have 
to  suppose  the  total  expansion  carried  out  in  a  series  of 
infinitely  small  steps,  each  being  represented  by  the  product  of 
the  existing  pressure  into  a  small  volume  increase  dv  and  then 
add  them  all  together.  Analytically  we  can  express  it  thus — 
Total  maximum  work  done  in  expansion  from  v±  to  v2 


•/: 


In  order  to  carry  out  this  integration  we  must  know  p  in  terms 
of  V'  If  the  system  is  gaseous  we  can  make  use  of  the  gas 
law,  viz.  — 

pv  =  «RT 

Where  n  is  the  number  of  gram-molecules  of  gas  in  volume  v, 
and  R  is  the  gas  constant.  Substituting  this  value  of  p  in  the 
integral  we  obtain  — 

Work  done  * 


=  /'  V  -  f^  =  *RT  P* 

l  ",        ''i  K  * 

=  «RT  log  —  (where  log  =  loge) 


The  maximum  work  done  when  one  gram-mole  of  gas  expands 
isothermally  and  reversibly  from  z^  to  z/2  is  therefore  — 

RTlog-2 
°"i 

Since  we  have  already  assumed  the  gas  laws  for  the  system 


12 


A    SYSTEM  OF  PHYSICAL    CHEMISTRY 


considered   we   can    evidently   write   this    expression   in   an 
alternative  form,  viz. — 

Maximum  work  per  gram-mole  =  RT  log  ~ 

A 

It  may  be  pointed  out,  parenthetically,  that  in  the  isothermal 
evaporation  of  a  liquid  we  have  a  case  in  which  there  may  be  a 
large  volume  change,  the  pressure  remaining  constant,  namely, 
the  pressure  of  the  saturated  vapour.  Suppose  a  cylinder  fitted 
with  a  piston  contains  some  liquid  and  saturated  vapour. 
Let  us  consider  what  is  the  value  of  the  maximum  work  which 
will  be  done  when  one  gram-mole 
of  the  liquid  is  vaporised.  We  sup- 
pose that  there  is  more  than  one 
gram-mole  of  liquid  present  to  start 

P\  with,  so  that  throughout  the  whole 

operation  the  vapour  is  saturated. 
Suppose  the  piston  (Fig.  39)  is  raised 
in  a  reversible  manner,  i.e.  infinitely 
slowly,  the  pressure  outside  being 
very  nearly  the  value  of  /,  the 
pressure  of  the  saturated  vapour 
(i.e.  the  pressure  outside  the  piston) 
is  p  —  dp.  The  process  is  assumed 
to  take  place  isothermally,  and  hence 
heat  must  be  continuously  supplied  from  the  outside  to  supply 
that  which  has  become  latent  in  the  process  of  vaporisation. 
If  the  increase  in  volume  of  the  vapour  corresponding  to  the 
production  of  one  gram  mole  of  vapour  is  V,  and  the  decrease 
in  the  volume  of  liquid  at  the  same  time  is  z>,  the  maximum 
external  work  done  by  the  system  as  a  whole  is/(V  —  v).  In 
general  we  can  neglect  v  compared  to  V.  The  work  is  then 
/V.  Further,  if  we  assume  that  the  vapour  obeys  the  gas 
laws,  we  can  write  maximum  external  work  done  in  vaporising 
one  gram-mole  of  gas  =/V  =  RT  where  R  is  1*985  calories 
and  T  the  absolute  temperature.  It  may  be  noted  that  the 
piston  may  start  from  any  level,  i.e.  from  the  surface  of  the 
liquid  itself  or  from  any  position  above,  for  we  only  deal  with 


=P-—  _ 


~mal  position 
of  Piston. 


Initial  position 
of  Piston. 


pIG§  29. 


OSMOTIC    WORK  13 

change  in  V,  not  with  the  initial  volume  actually  possessed  by 
the  system.  The  student  will  note  that  it  has  not  been 
necessary  to  take  into  consideration  the  actual  value  of  the 
heat  quantity  added.  One  must  be  careful,  however,  not  to 
imagine  that  the  heat  added  and  the  external  work  done  are 
identical  in  this  case.  In  fact,  the  heat  added  is  very  much 
greater  than  the  heat  equivalent  of  the  external  work  done. 
There  has  been  a  considerable  change  in  the  internal  energy 
of  the  gram-mole  of  liquid  due  to  its  vaporisation.  In  the 
case  of  water,  for  example,  the  heat  required  to  vaporise  one 
gram  at  100°  C.  is  about  540  calories.  The  external  work 
done  (A)  against  the  atmosphere  is  about  40  calories,  the  so- 
called  "external  latent  heat,"  leaving  500  calories  as  the 
"internal  latent  heat."  This  term  represents  the  change  in  U 
which  has  taken  place  in  the  transforming  of  one  gram  of 
liquid  water  into  water  vapour.  For  the  whole  process — 

U  =  A  — Q     or    Q  =  A  — U. 

In  the  above  we  have,  however,  only  been  dealing  with  the 
magnitude  of  the  single  term  A. 

MAXIMUM  OSMOTIC  WORK  IN  SOLUTIONS. 

We  are  already  acquainted  with  the  fact  that  a  solution, 
such  as  a  dilute  sugar  solution,  obeys  the  gas  laws.  Let  us 
regard  this  as  an  experimental  fact — as  it  has  indeed  been 
shown  to  be.  Now,  the  process  of  diluting  a  solution  is  a 
familiar  one,  and  we  can  see  that,  under  certain  conditions, 
this  is  analogous  to  diluting  a  gas,  i.e.  increasing  the  volume  of 
the  gas.  It  seems  justifiable,  therefore,  to  conceive  of  work 
being  done  by  the  solution,  or  rather  by  the  solute  in  the 
solution  during  dilution,  for  we  have  here  the  two  necessary 
factors,  pressure  (i.e.  osmotic  pressure)  and  volume  change. 
In  order  to  make  the  dilution  process  as  mechanical  as 
possible,  and  thereby  bring  out  the  close  analogy  to  the 
expansion  of  a  gas,  we  have  recourse  to  a  device  called  the 
semi-permeable  membrane,  with  which  we  are  already  familiar 
in  our  previous  discussion  of  solutions.  The  semi-permeable 


14         A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

membrane  allows  solvent  to  pass  freely  through  it,  but  not  the 
solute,  i.e.  the  solute  can  exert  its  osmotic  pressure  against 
the  semi-permeable  membrane.  For  our  present  purpose  of 
calculating  maximum  work  it  does  not  matter  in  the  least 
whether  we  can  realise  in  practice  such  a  membrane  or  not, 
for  we  are  considering  a  limiting  case  in  either  event.  As  a 
matter  of  fact,  however,  several  semi-permeable  membranes 
(or  practically  semi-permeable  membranes)  have  been  used 
in  practice,  notably  that  consisting  of  copper  ferrocyanide, 
deposited  in  the  walls  of  a  porous  pot  which,  while  it  allows 
water  to  pass  freely  through  it,  altogether  prevents  the  passage 
of  sugar  dissolved  in  the  water. 


Solvent 

Solvent 

r$£f. 

W  eight  s- 

•^Tx, 

Semipermeable 
membrane  ' 

i#&W?&$ 
Solution 

Solution 

Semipermeable 


FIG.  40. 

Suppose,  now,  we  had  a  quantity  of  solution  in  a  cylinder, 
and  above  the  solution  a  layer  of  pure  solvent,  a  semi- 
permeable  membrane  being  placed  between  the  two  layers 
just  where  the  solution  meets  the  solvent  (see  Fig.  40).  To 
keep  the  semi-permeable  membrane  motionless,  we  must 
imagine  weights  placed  upon  it  so  as  to  just  balance  the 
osmotic  pressure  of  the  solute  in  the  solution  which  is  tending 
to  press  the  membrane  out  in  exactly  the  same  manner  as  a 
gas  tends  to  push  an  ////permeable  piston  out.  To  obtain  the 
maximum  work  which  could  be  done  by  diluting  the  system 
from  volume  V1  to  volume  V2,  that  is,  in  ordinary  chemical 
nomenclature  from  "  dilution  "  V1  to  "  dilution  "  V2,  we  suppose 
the  semi-permeable  membrane  or  piston  to  rise  infinitely  slowly, 
the  pressure  due  to  the  weights  being  adjusted  so  as  to  be  less 


MAXIMUM  OSMOTIC    WORK  15 

than  the  osmotic  pressure  by  the  infinitely  small  quantity  dP. 
We  must  imagine  the  weights  continually  to  decrease  in 
number  so  as  to  keep  pace  with  the  continuously  decreasing 
osmotic  pressure  of  the  solution,  for  of  course  the  osmotic 
pressure  of  the  solution  decreases  as  the  dilution  increases. 

As  the  membrane  moves  up,  solvent  passes  through  it,  no 
pressure  difference  being  set  up  on  the  two  sides  of  the 
membrane,  so  far  as  the  solvent  is  concerned,  for  the  mem- 
brane js  perfectly  permeable  to  the  solvent  although  im- 
permeable to  the  solute.  It  may  be  pointed  out  that  if 
no  weights,  i.e.  no  opposing  pressure,  had  been  placed  upon 
the  membrane,  the  latter  would  move  instantaneously  up 
through  the  solvent,  no  work  being  done  thereby ;  just  as  we 
saw  no  work  was  done  when  a  gas  expanded  into  a  vacuum. 
We  are  at  present  dealing,  however,  with  the  exactly  opposite 
limit,  namely,  the  production  of  maximum; work.  As  in  the 
case  of  the  gas,  the  maximum  work  for  the  whole  volume  or 
dilution  increase  is  given  by  the  expression — 


where  P  denotes  the  osmotic  pressure  of  the  solution.  To 
evaluate  this  integral  we  must  know  P  in  terms  of  the  dilution. 
This  is  given  by  the  experimental  fact  that  osmotic  pressure 
follows  the  gas  laws,  viz. — 

P  V  =  RT     or     P  =  -  r  =RTC 

when  C  is  the  reciprocal  of  dilution  and  represents  the 
concentration  of  the  solute  in  the  solution  C  =  v  To  be 

able  to  give  a  definite  numerical  value  to  R  we  must  deal 
with  a  certain  mass  of  solute,  say  i  gram-mole,  in  which 
case  R=  1*985  calories.  The  maximum  osmotic  work  done 
by  the  system  in  diluting  i  gram-mole  of  solute  from  a 
dilution  Vx  to  a  dilution  V2  is  given  by  the  expression — 

v*d~V  V 

RT  |       -  =  RT  log  rf 
V  V  i 


1 6         A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

If  we  dealt  with  n  gram-moles  of  solute  the  maximum 
osmotic  work  would  be  n  times  as  great,  viz.— 

"«RT  log  ^2 

Using  osmotic  pressure  terms  instead  of  volume  terms,  since 

RT 
we  know  the   relation   between  them,  namely,  P  =  -  ^-,  the 

above  expression  can  be  written  for  the  case  of  one  gram 
mole  as — 

Maximum  osmotic  work  =  RT  log  =^ 
or  since  V  =  -^, 

r* 

Maximum  osmotic  work  =  RT  log  ~ 

Where  Cj  is  the  initial  concentration  of  the  solution  and  C2  its 
final  concentration.  It  is  well  to  notice,  however,  that  we 
have  employed  the  gas  law  expression  in  dealing  with  these 
solutions.  The  accuracy  of  the  expression  is,  therefore, 
limited  by  the  range  of  applicability  of  the  gas  laws  to  the 
solutions.  That  is,  the  above  expression  gives  the  maximum 
osmotic  work  for  dilute  solutions  only. 

EXTERNAL  WORK  DEFINED  AS  A  CHANGE  IN  FREE  ENERGY 
OF  THE  SYSTEM  AND  SURROUNDINGS. 

When  a  body  does  work  at  constant  temperature  it  uses  up 
energy  either  at  its  own  expense  or  at  the  expense  of  its 
surroundings.  This  energy  is  called  the  free  or  available 
energy,  to  distinguish  it  from  the  quantity  which  we  have 
denoted  by  the  term  U,  which  is  the  change  in  internal  or  total 
energy.  Free  energy  can  be  denoted  by  A,  since  it  is  exactly 
equal  to  the  maximum  work  done  in  isothermal  changes ^  The 

1  In  an  irreversible  process,  when  maximum  work  is  not  done,  there  is  a 
decrease  in  the  free  energy,  and  this  decrease  is  greater  than  the  work  done. 
Some  of  the  free  energy  has  been  simply  lost  or  dissipated  as  heat.  It  is 
only  in  a  reversible  process  that  the  free  energy  is  entirely  converted  into 


FREE   OR  AVAILABLE  ENERGY  17 

relation  between  the  free  and  total  energy  of  a  system  is  by  no 
means  an  obvious  one,  in  fact  we  cannot  attempt  at  this  stage 
to  show  the  relation  between  the  two  until  we  have  discussed 
the  second  law  of  thermodynamics,  the  relation  between  the 
two  being  given  by  the  expression  known  as  the  Gibbs- 
Helmholtz  equation.  The  only  point  emphasised  here  is  that 
the  total  or  internal  energy  change  is  not  the  same  thing  as 
the  free  or  available  energy  change.  The  use  of  the  term 
"  total  "  in  regard  to  U  must  not  suggest  to  the  mind  that  the 
free  energy  is  a  fraction  of  this  only.  As  a  matter  of  fact,  the 
free  energy  decrease  (i.e.  the  maximum  work  output  at  constant 
temperature)  in  some  processes  actually  exceeds  the  decrease 
in  total  energy  (U)  taking  place  in  the  same  process,  and  in 
some  extreme  cases  the  total  energy  of  the  system  may  actually 
increase  whilst  at  the  same  time  external  work  is  done  by  the 
system  (i.e.  free  or  available  energy  is  given  up,  e.g.  the  process 
of  vaporisation).  The  possibility  of  this  arises  from  the  fact 
that  the  change  in  total  energy  has  its  origin  in  the  system  itself, 
whilst  the  free  energy,  on  the  other  hand,  may  be  due  partly 
to  the  system  itself  and  partly  or  wholly  to  the  surroundings. 
As  a  matter  of  fact,  in  the  case  of  an  isothermal  gas  expansion, 
or  that  of  osmotic  work  in  a  dilute  solution  such  as  we  have 
been  considering,  free  energy  which  manifests  itself  as  external 
work  is  ultimately  drawn  from  the  surroundings  entirely,  i.e. 
from  the  heat  content  of  the  surroundings.  In  processes 
occurring  in  other  systems,  such  as  liquids,  the  free  energy  may 
be  partly  drawn  from  the  resources  of  the  system  itself  as  well 
as  from  the  surroundings.  It  might  be  thought  that  there  is  so 
much  vagueness  about  the  sources  of  free  energy  that  no 
relation  could  be  established  between  it  and  the  U  of  the 
system.  We  shall  see  later,  however,  that  if  we  restrict 
ourselves  in  any  system  whatsoever  to  reversible  processes, 
that  is,  to  cases  in  which  maximum  work  is  done  and,  therefore, 
all  the  free  energy  is  given  out  in  the  form  of  work  at  constant 

external  work,  this  work  being,  therefore,  the  maximum  of  which  the 
system  is  capable.  Further  note  that  all  this  refers  to  changes  at  constant 
temperature  only.  The  free  energy  of  a  system  alters  with  temperature, 
whether  work  (maximum  or  otherwise)  is  done  or  not. 

T.C.P. — II.  C 


i8         A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

temperature,  there  is  a  relationship  between  A  and  U.  As 
already  mentioned,  this  is  expressed  in  the  Gibbs-Helmholz 
equation  to  which  we  shall  come  later.  For  the  sake  of 
brevity  it  is  customary  to  use  the  expression,  "  the  free  energy 
of  a  system,"  to  denote  the  free  or  available  energy  possessed 
by  the  system  and  its  surroundings  under  the  given  conditions. 
A  system  is  said  to  "possess  "  free  energy  whether  it  is  made 
to  do  work  or  not,  such  free  energy  content  being  measured 
by  the  maximum  work  the  system  could  perform  under  the 
given  conditions. 

THE  MAXIMUM  WORK  (OF  FREE  ENERGY  CHANGE)  INVOLVED 
IN  A|  (REVERSIBLE)  ISOTHERMAL  PROCESS  WHICH  TAKES 
PLACE  IN  MORE  THAN  ONE  STAGE. 

In  the  cases  which  we  have  been  considering,  namely,  the 
expansion  of  a  gas  or  the  dilution  of  a  solution,  the  whole 
operation  took  place  in  one  stage  or  step,  i.e.  the  piston  or 
membrane  simply  moved  through  a  certain  distance  under 
given  pressure  relations.  We  now  want  to  consider  a  slightly 
more  complicated  process,  and  we  shall  consider  that  this  is 
also  of  the  reversible  kind,  i.e.  each  stage  must  be  so  carried 
out  as  to  produce  maximum  work.  We  shall  confine  our 
attention  to  the  following  three-stage  process,  as  this  is  a  very 
typical  one  which  occurs  again  and  again  in  the  thermody- 
namical  treatment  of  physical  and  chemical  processes.  Let  us 
suppose  that  in  a  cylinder,  I  (Fig.  41),  we  have  a  liquid  (say 
water)  having  a  certain  vapour  pressure  /0  at  a  given 
temperature.  In  the  second  cylinder,  II,  we  have  a  solution, 
the  solvent  being  identical  with  the  pure  liquid  in  I,  the  solute 
being  a  non-volatile  one.  The  vapour  pressure  of  the  solution, 
i.e.  the  pressure  of  the  vapour  of  the  solvent,  is  p±  where  pl  is 
less  than  /0,  since  the  presence  of  the  solute  lowers  the  vapour 
pressure  of  the  solvent.  The  solution  in  II  is  at  the  same 
temperature  as  the  liquid  in  I.  The  quantity  of  solution  in  II 
is  so  great  that  the  addition  of  one  gram-mole  of  solvent  is 
supposed  not  to  have  any  measureable  diluting  effect,  i.e.  the 
concentration  of  the  solute  in  II  is  supposed  to  remain 
constant.  The  problem  before  us  is  this  : — 


A    THREE-STAGE   ISOTHERMAL  PROCESS       19 

What  is  the  maximum  work  involved  in  the  process  of 
transferring  one  gram-mole  of  the  liquid  from  I  and  adding  it 
to  the  solution  in  II,  the  temperature  throughout  remaining 
constant  and  the  concentration  of  II  being  likewise  constant? 
In  other  words,  what  is  the  maximum  work  done  in  isother- 
mally  distilling  one  gram-mole  of  liquid  from  1  into  the 
solution  in  II  ?  In  the  first  place,  it  will  not  do  simply  to  place 
some  of  the  pure  solvent  (i.e.  the  liquid  in  I)  directly  in 
contact  with  the  solution  in  II,  for  this  process  would  entail 
no  work  at  all,  since  the  solute  would  spread  itself  out  into  the 
layer  of  solvent  against  zero  osmotic  pressure.  The  same  idea 


/ 

1 

1 
1 

ffinc 
/of/- 
/Inn 

'of 

Vapour 

—     Vapour  —  >>• 

\ 

la               Ib 

v  position 
fetai 

ial  position                             1 
Alston. 

el  of  Liquid 
face. 

i 
Yappur 

T*pbu7- 

of  Piston. 

,  Final  posrt/cn 
•  of  Piston 

Vapour 

Vapzxj  r 

\Lev 
sur 

^LiquTcT- 

'''/  //'//// 
'///'///, 

.Solution' 

W^ 

\>i/*/  of 
Solution 

n 


FIG.  41. 


may  be  expressed  by  saying  that  neither  will  it  do  to  insert  a 
weightless  semi-permeable  membrane  on  the  surface  of  the 
solution  with  solvent  above  it,  and  allow  this  membrane  to  be 
pressed  through  the  solvent  by  the  osmotic  pressure  of  the 
solute,  as  here  again  no  work  would  be  done,  owing  to  the 
fact  that  no  pressure  resistance  would  be  offered  to  the  osmotic 
pressure.  To  obtain  the  maximum  work  the  whole  process 
must  be  carried  out  in  a  reversible  manner,  that  is  to  say,  in 
each  single  stage  considered  the  difference  of  pressure  on  the 
sides  of  the  piston  or  membrane  must  never  be  finite,  i.e.  must 
never  exceed  a  value  represented  by  dp.  To  carry  out  the 
whole  process  reversibly  and  thereby  obtain  the  maximum 


20         A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

output  of  work,  we  must  consider  it  in  three  successive  stages 
(ff.  Fig.  41),  viz.— 

(1)  One  gram-moie    of   liquid  in   I  is   isothermally   and 
reversibly  vaporised  at  the  pressure  /0. 

(2)  The  gram-mole  of  vapour  is  now  isolated  by  means  of 
an  impermeable  shutter  pushed  in  over  the  liquid,  no  work 
being  entailed  thereby.     That  is,  we  now  have  one  gram-mole 
of  vapour  in  a  small  cylinder  la  at  pressure  /0.     We  suppose 
now  that  this  is  allowed  to  expand  isothermally  and  reversibly 
until  its  pressure  has  fallen  to  p±  +  dP> tnat  is>  practically  to  plt 
the  volume  now  being  z>1}  say.     That  is,  la  has  become  I&. 

(3)  The  cylinder  I&  is  brought  into  contact  with  II,  and 
since  the   pressure   is   practically  identical  with  that  of  the 
vapour  in!  II,  the  gram-mole  of  vapour  may  be  added  to  II 
reversibly.     As  it  is  pressed  in  at  a  pressure  p±  +  <^>  naturally 
it  causes  condensation  of  an  equal  mass  of  vapour,  and  if  the 
process  is  carried  out  infinitely  slowly  the  temperature  will 
remain  constant  (heat  being  passed  out  to  the  surroundings). 
The  operation  of  transferring  one  gram-mole  of  liquid  in  I  to 
solution  in  II  is  now  complete,  and  has  been   carried   out 
reversibly.     By  adding   together  the  various  work   terms  we 
can  obtain  at  once  the  maximum  work  involved  in  the  process 
as  a  whole.     The  work  terms  are  : 

In  operation  (i)  an  amount  of  work  pQvQ  is  done  by  the 
system  (VQ  denoting  the  volume  of  one  mole  of  saturated 
vapour  in  I).  Since  the  vapour  remains  saturated  the  pres- 
sure remains  constant.  We  have  neglected  the  molecular 
volume  of  liquid  as  compared  to  the  molecular  volume  of 
vapour.  In  operation  (2),  the  sliding  of  a  frictionless,  weight- 
less shutter  involves  no  work.  That  is,  the  act  of  isolat- 
ing the  gram-molecule  of  vapour  in  la  involves  no  work,  but 
work  is  involved  when  we  allow  this  to  expand  reversibly 
against  an  opposing,  continuously  decreasing,  external  pressure 
which  only  differs  from  that  inside  the  cylinder  by  the  amount 
dp.  Assuming  the  gas  law  for  the  vapour,  the  work  done  by  the 
vapour  is — 

RT  lo^ 


A    THREE-STAGE   ISOTHERMAL   PROCESS        21 

The  vapour  is  now  in  the  state  denoted  by  I&.  In  opera- 
tion (3)  we  have  the  reverse  kind  of  process  to  that  in  opera- 
tion (i).  In  this  operation  the  volume  of  the  vapour  decreases, 
i.e.  work  is  done  by  the  surroundings  upon  the  system,  the 
amount  being  —p-^v^  the  negative  sign  denoting  that  there 
is  a  decrease  in  volume  as  condensation  is  taking  place.  We 
here  neglect  the  increase  in  the  volume  of  the  solution  owing 
to  this  condensation  compared  to  vt.  These  are  all  the  work 
terms  involved,  and  hence  for  the  total  process  the  maximum 
work  done — 

=  maximum   work)   .   (maximum   work)   ,   (maximum   work 
in  operation  (i)$      1  in  operation  (2)3  ~*~\  in  operation  (3) 

That  is— 

Maximum  work  in  process  =  A  —pQVQ  +  RT  log  --  —  p&\ 

VQ 

Now  it  will  be  observed  that  we  have   assumed  the  vapour 

obeys  the  gas  laws,  since  the  middle  term  is  the  integral  I    pdv 

J  v0 

RT 

evaluated  by  putting/  =  —  - 

If  the  assumption  of  the  gas  law  is  justifiable  in  regard  to  this 
term,  it  must  be  likewise  justifiable  in  regard  to  the  first  and 
last  terms.  The  first  term  is  /Oz/0  or  RT.  The  last  term  is 
p-fli  or  RT,  for  the  temperature  is  the  same  in  both  cases,  and 
R  is  likewise  the  same  since  we  have  considered  the  same 
mass  (one  gram-mole)  of  vapour  throughout.  That  is,  the  first 
and  last  terms  cancel,  and  we  get — 

#j 

Total  maximum  work  in  process  =  A  =  RT  log  — 

This  three-stage  process  has  therefore  turned  out  to  be  iden- 
tical with  a  single-stage  one  (namely,  the  stage  ln  to  I&).  This 
simple  result  is,  however,  only  obtained  by  the  assumption 
that  the  vapour  obeys  the  gas  laws. 

Let  us  now  consider  the  same  process^  but  no  longer  assume 
the  applicability  of  the  gas  law  pv  =  RT. 


22         A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

Operation  (i).  —  The  work  done   by  the  system   is  again 
/oz/0'  but  this  is  no  longer  necessarily  equal  to  RT. 

fvi 
Operation  (2).  —  The  work  done  is  /    pdi\  but  we  cannot 

J   VQ 

write  this  in   the  form  RT  log  ~  since  the  gas  law  is  not 

assumed. 

Operation  (3).  —  The  work  done  is  —  A7'i>  which  again  is 
not  necessarily  equal  to  RT. 

Hence  the  total  maximum  work  A  is  given  by  — 

A  = 

(vl 
The  term  /   pdv  may  be  integrated  by  parts  whatever  the 

J  VQ 

relation  between  p  and  v  may  be.     That  is,  we  can  always 
write  — 

£1  [PI 

pdv  =p1vi  —  PQVQ  —  \    vdp 
o  "  Po 

Hence  — 

A  =    V 


PI 
+A?'i  —  A>7'o  —  /    vdp  — 

J  Po 

epi 

=  —  I    vdp 

J 


or  A 

J  Po 

This  expression  holds  for  the  process  involving  the  whole  of 
the  three  stages  whether  the  gas  laws  hold  for  the  vapoiir  or  not. 
This  is  therefore  the  most  general  term  for  the  maximum 
work  done  in  a  three-stage  isothermal  process  of  the  type  con- 
sidered. If  we  find  as  a  special  case  that  the  gas  laws  happen 
to  be  obeyed  by  the  vapour  in  question  sufficiently  well,  we 

/PI  RT 

vdp  by  writing    v  —  - 
PO  P 

which  thus  yields  — 

A  =  —  RT  log^1  =  RT  log^°  =  RT  log  ^ 
6/o  B/i  "o 

as  before. 

It  must  be  remembered  that  any  single  work  term  of  the 


REVERSIBILITY  AND   IRREVERSIBILITY        23 

type  considered  must  be  the  product  of  a  pressure  into  an 
increase  in  volume,  i.e.  pQvQ  or  p-^v^  or pdv  or  jpdv.  One  can- 
not represent  a  single  work  term  by  the  product  of  volume 
into  increase  in  pressure,  say  vdp  or  jvdp.  It  happens,  how- 
ever, that  the  three  consecutive  work  terms  involved  in  the 
process  considered  above  do  reduce  down  to  a  single  vdp  term 
(owing  to  the  operation  of  integration  by  parts),  and  this 
result,  namely — 


[PI  fPo 

=  —  /    vdp=\    vdp 

J PO  J  PI 


is  reached  whether  the  system  or  substance  considered  obeys 
the  gas  laws  or  not.  Although  physically  speaking  we  can- 
not represent  a  single  work  term  by  vdpt  it  is  evident  that  if 
we  are  studying  the  special  case  of  the  work  of  expansion  at 
constant  temperature  of  a  vapour  or  gas  which  obeys  the  gas 
laws,  the  work  done  —  pdv,  and  this  is  numerically  equal  to 
—  vdp  because  the  expression  d(pv)  =  vdp  -{-pdv  =  o,  since 
pv  is  a  constant  at  constant  temperature  by  Boyle's  Law.  It  is 
only,  however,  when  d(pv)  =  o,  that  is,  when  the  gas  laws  are 
obeyed,  that  we  can  interchange  pdv  and  vdp  terms.  When 
the  system  does  not  obey  the  gas  laws  it  is  necessary  to  find 
some  relation  between  p  and  v  which  will  allow  us  to  evaluate 
any  integrals.  For  vapours  one  may  apply  as  a  first  approxi- 
mation the  van  der  Waals  equation  for  this  purpose,  treating  a 
and  b  as  constants. 

THE  DISTINCTION  BETWEEN  THE  NATURAL  OR  SPONTANEOUS, 
AND  THEREFORE  IRREVERSIBLE  METHOD  OF  CARRYING 
OUT  A  REACTION  AND  THE  THERMODYNAMIC  REVERSIBLE 
METHOD  OF  CARRYING  OUT  THE  SAME  REACTION. 

The  distinction  is  best  made  clear  by  means  of  an 
example.  Take  the  case  of  the  chemical  reaction  which 
occurs  between  water  and  sulphuric  acid.  Let  us  think  of 
an  apparatus  similar  to  that  indicated  in  Fig.  42.  In  one 
vessel,  A,  there  is  a  quantity  of  liquid  water,  and  in  contact 
with  it  some  saturated  vapour  at  pressure  /0.  The  vapour 
fills  the  space  on  the  left-hand  side  of  the  tap  C.  In  the 


A    SYSTEM  OF  PHYSICAL   CHEMISTRY 


^ 


vessel  B  there  is  some  concentrated  sulphuric  acid,  that  is  acid 
containing  a  little  water,  and  above  this  acid  is  some  vapour 

in  equilibrium  with  the 
water  in  this  sulphuric 
acid  mixture.  The  partial 
pressure  of  the  water  vapour 
is  here  /',  where  /'  is  much 
less  than  pQ.  This  water 
vapour  at  low  pressure 
(along  with  some  sulphuric 
acid  vapour  which  does  not 


Po 


J 


B 


FIG.  42. 


come  into  the  calculation) 
occupies  the  space  on  the 
right  of  the  tap  C.  If  we  simply  open  the  tap,  water  vapour 
would  stream  from  left  to  right,  that  is  from  the  region  of  high 
pressure  /0  to  that  of  low  pressure  /'.  If  a  piston  were  placed 
in  the  tube  it  would  be  driven  at  a  speed  not  by  any  means 
infinitely  slowly,  and  the  pressure  difference  on  the  two  sides 
of  the  piston  would  be  finite,  i.e.  (pQ  — /').  This  process,  which 
is  the  spontaneous  one,  is  an  irreversible  one,  since  the  piston 
is  not  made  to  move  infinitely  slowly  with  infinitely  small 
pressure  difference  on  the  two  sides. 

We  can  carry  out  the  same  transfer  of  water  to  concentrated 
sulphuric  acid  reversibly,  however,  by  following  out  the  three- 
stage  work  process  already  described.  First,  a  certain  quantity 
of  water  is  vaporised  by  pulling  out  a  piston  infinitely  slowly, 
the  pressure  on  the  outside  of  the  piston  being /0  —  dp.  This 
vapour  is  now  isolated  and  expanded,  doing  work  against  a 
continuously  decreasing  external  pressure,  which  never  differs 
from  the  actual  pressure  of  the  vapour  by  more  than  the 
infinitely  small  quantity  dp.  In  this  way  the  pressure  of  the 
water  vapour  is  brought  to  /',  or  rather  to  (pr  -f-  dp),  and  is 
then  introduced  into  the  space  over  the  concentrated  sulphuric 
acid  and  pressed  in,  i.e.  condensed  infinitely  slowly  owing  to 
the  pressure  difference  on  the  two  sides  of  the  piston  being 
infinitely  small.  The  total  mechanical  work  which  is  here 


maximum  work  is,  as  we  have  seen,  —  I     rdp 

J  P* 


KIRCHHOFF^S  EQUATION  25 

It  will  thus  be  clear  that  we  can  carry  out  the  same 
process  either  in  the  natural  or  irreversible  way,  in  which  case 
we  do  not  get  a  maximum  output  of  work,  or,  on  the  other 
hand,  we  can  imagine  the  process  carried  out  reversibly  in  the 
thermodynamic  sense,  in  which  case  we  do  get  a  definite 
maximum  quantity  of  work,  expressible  in  terms  of/  and  v. 

KIRCHHOFF'S  EQUATION  FOR  HEAT  OF  REACTION  (WHICH 
INVOLVES  THE  PRINCIPLE  OF  THE  FlRST  LAW  OF 
THERMODYNAMICS). 

Suppose  that  a  system  changes  from  a  state  A  to  a  state  B,  i.e. 
as  a  particular  case,  let  us  consider  a  chemical  reaction  of  any 
kind  whatsoever  in  which  at  a  temperature  T,  a  quantity  of  heat 


777, 

hdt 
A 

V    a  tj 

^tidt 

T 

I 

^ 

<?  >                    ^ 

FIG.  43. 

Qv  is  absorbed  in  the  transformation  of  reactants  into  resultants 
at  constant  volume,  then  (—  Qw  =  U).  Suppose  that  the  same 
reaction  goes  at  a  higher  temperature,  T  -f-  dTT,  then  the  heat 
which  will  be  absorbed  may  be  represented  by  Q«  +  ^Q*>. 
Further,  let  us  denote  by  h  the  heat  capacity  of  the  reactants 
and  by  h'  the  heat  capacity  of  the  resultants.  We  can  go 
from  the  stage  A  to  the  stage  B  by  two  independent  paths, 
indicated  in  the  diagram  (Fig.  43),  namely,  AmE  and  A//B. 
The  point  A  denotes  the  system  consisting  entirely  of  the 
reactants  and  at  the  temperature  T ;  the  point  B  denotes  the 
system  consisting  entirely  of  resultants  at  temperature  T  -f-  dTT. 
By  the  First  Law  of  Thermodynamics,  i.e.  by  the  principle  of 
the  conservation  of  energy,  the  net  heat  absorption  or  evolution 
via  A;/B  must  be  the  same  as  via  Am~B,  since  the  system  starts 
from  the  point  A  and  ends  in  both  cases  at  the  point  B,  without 
any  volume  change,  so  that  the  heat  change  is  identical  with 


26         A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

the  internal  energy  change.  Going  via  AwB  we  start  with  the 
reactants  at  A  and  raise  their  temperature  to  T  -}-  ^T,  an 
amount  of  heat  -f  //<//  Ueing  thereby  absorbed  (the  positive 
sign  denotes  heat  absorbed).  Having  now  reached  /w,  we 
suppose  the  reaction  to  take  place  (at  T-f-^T)  whereby  the 
reactants  are  converted  into  resultants,  the  heat  absorbed  being 
Q»  ~h  dQv  We  have  now  reached  B.  Let  us  start  again  at  A, 
and  allow  the  reaction  to  go  at  T,  the  heat  absorbed  is  QB.  We 
have  now  reached  n.  Now  raise  the  temperature  of  the 
resultants  to  T  -f  ^T,  the  heat  absorbed  being  -f  //<//,  we  are 
again  at  B.  Equating  the  heat  effects  by  the  two  paths  we 
obtain — 

-f  hdt  -f  Q»  +  JQV  =  Q*  +  h'dt 

or  *^=K-h  =  Kv-h* 

at 

where  the  suffix  v  is  introduced  to  denote  constant  volume ; 
or  *£  =  *,-#, 

This  is  KirchhofT's  equation.  Its  great  importance  lies  in 
the  fact  that  it  allows  one  to  calculate  the  temperature  co- 
efficient of  heat  effects  from  measurements  of  the  specific  heats 
of  the  substances  involved  in  a  very  much  more  accurate 
manner  than  could  possibly  be  done  by  actually  measuring  Q 
or  Qw  at  two  different  temperatures. 

THE  WORK  DONE  BY  THE  SURROUNDINGS  (i.e.  THE  EXTERNAL 
AGENCY)  IN  COMPRESSING  A  GAS  (a)  ISOTHERMALLY, 
(b)  ADIABATICALLY. 

During  the  process  of  isothermal  compression,  the  heat 
which  is  produced  and  evolved  is  absorbed  by  the  surroundings 
which  are  supposed  to  be  of  so  great  extent  that  the  temperature 
of  the  system  remains  constant,  provided,  of  course,  that  the 
compression  takes  place  sufficiently  slowly  to  allow  of  the  heat 
being  conducted  away.  In  an  adiabatic  compression,  on  the 
other  hand,  we  suppose  the  gas  to  be  isolated  in  a  "  heat- 
tight  "  case  so  that  no  heat  can  be  added  to  or  abstracted 


DEFINITION  OF   THE   PERFECT  GAS 


27 


from  the  gas.  If  we  compress  this  system,  the  temperature 
rises,  and  this  rise  in  temperature  tends  to  make  the  gas 
expand,  i.e.  the  temperature  effect  opposes  the  compressing 
force,  and  hence  this  external  agency  will  have  to  do  more  work 
upon  the  gas  to  compress  it  to  the  final  volume  than  it  had  to 
do  when  the  gas  was  being  compressed  isothermally — the 
initial  and  final  volumes  being  the  same  in  each  case.  If  we 
make  use  of  a  pv  diagram,  the  area  represents  work  done,  and 
if  in  the  case  considered 
the  initial  volume  is  vlt 
and  the  final  volume  after 
compression  is  £'0,  then 
work  done  in  the  isothermal 
compression  can  be  repre- 
sented by  the  area  ABz^i 
(Fig.  44).  The  line  AB  is 
called  an  isothermal  line. 
In  order  to  indicate  a 
greater  work  output,  as  is 
the  case  in  the  adiabatic 
compression  between  the 
two  volume  limits 


FIG.  44. 

and  7'0,  we  must  trace  a  line  such  as  AC, 
the  adiabatic  work  being  represented  by  the  area  ACvQz\. 
The  line  AC  is  called  an  adiabatic  line.  On  traversing  AB 
the  temperature  of  the  gas  remained  constant.  On  traversing 
AC  the  temperature  of  the  gas  rose.  //  is  clear  that  an  adiabatic 
line  or  curve  is  steeper  than  the  isothermal  line  or  curve  on  the 
pv  diagram.  The  relative  position  of  the  two  lines  or  curves 
is  of  some  importance  when  we  come  later  to  study  thermo- 
dynamical  cycles. 


THERMODYNAMICAL  DEFINITION  OF  A  PERFECT  GAS. 

An  experiment  carried  out  by  Gay-Lussac,  and  afterwards 
improved  by  Joule,  can  be  briefly  outlined  as  follows  (cf.  Fig. 
45).  A  large  vessel  I  containing  gas  was  connected  by  means 
of  a  tap  to  a  similar  vessel  II,  which  had  been  evacuated.  The 
whole  was  immersed  in  a  bath,  the  temperature  of  which  was 


28         A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

measured  as  accurately  as  possible.  When  the  tap  was  opened 
gas  rushed  over  from  I  into  II.  There  is  thus  an  expansion  of 
gas^in  vessel  I,  and  the  gas  in  I  becomes 
colder  because  it  has  to  compress  the 
first  portions  of  the  gas  which  have  gone 
over  by  the  following  portions.  At  the 
same  time  the  temperature  in  II  rose 
because  the  first  portions  of  the  gas  were 


FlG     .  compressed  by  the  following   portions. 

The  temperature  of  the  bath  remained 

unchanged  as  far  as  Joule  could  observe  during  the  whole 
operation.  That  is,  the  cooling  effect  in  I  must  have  just 
compensated  the  warming  effect  in  II.  The  gas  as  a  whole 
did  not  rise  or  fall  in  temperature  on  expanding  into  a 
vacuum.  That  is  to  say,  on  the  whole  no  heat  was  either 
taken  in  or  given  out  by  the  gas  in  the  operation.  Now  for 
any  process  we  have  the  three  quantities  U,  Q,  and  A,  con- 
nected on  the  basis  of  the  First  Law  of  Thermodynamics  by 
the  relation  — 


In  the  above  operation  Q  =  o  (experimental  result).  Further, 
we  see  that  the  external  work  performed  by  the  gas  as  a  whole 
is  zero  in  expanding  from  I  to  II  (which  was  in  the  first  place 
evacuated).  The  whole  system  is  I  -}-  II  and  the  volume  of 
I  -}-  II  is  constant,  hence  no  external  work  is  done  by  the 
passage  of  the  gas.  This  holds  good  even  if  there  had  been 
some  gas  present  in  II  to  start  with,  for  this  condition  is 
evidently  realised  when  passage  of  gas  has  begun.  That  is, 
A  =  o.  Hence  from  the  First  Law  it  follows  that  U  =  o. 
That  is  to  say  that  the  total  or  internal  energy  of  the  gas  has 
not  changed.  (U  represents  change  in  internal  energy  not 
internal  energy  itself.)  We  therefore  arrive  at  the  following 
important  conclusion  :  — 

The  internal  energy  of  a  given  mass  of  gas  is  independent 
of  the  volume  occupied.  As  a  matter  of  fact  (as  shown  by  the 
porous  plug  experiment,  which  we  will  consider  later),  any 
actual  gas  only  approximates  to  this  statement.  There  really 


THE    THERMODYNAMICAL    CYCLE  29 

was  a  very  slight  change  in  temperature  in  the  bath  in  the 
Gay-Lussac- Joule  experiment,  though  the  methods  employed 
were  not  sufficiently  delicate  to  indicate  it.  This  slight  change 
in  temperature  has  been  traced  to  the  work  done  by  the 
molecules  in  drawing  themselves  apart  against  their  mutual 
attractive  forces.  A  perfect  gas  from  the  kinetic  standpoint 
is  one  in  which  no  attractive  forces  exist.  In  the  case  of  a 
perfect  gas,  therefore,  there  would  be  absolutely  no  change  in 
temperature  if  it  were  to  go  through  the  process  described.  In 
other  words,  there  would  be  no  change  in  the  internal  energy, 
i.e.  U  =  o.  We  can  thus  define  a  perfect  gas  as  a  substance,  the 
internal  or  total  energy  of  a  given  mass  of  which  is  independent 
of  the  •volume  occttpied  by  that  mass. 

The  only  other  factor  upon  which  internal  energy  (U) 
depends  is  that  of  temperature.  We  can  thus  as  alternatively 
define  a  perfect  gas  as  a  substance  the  internal  energy  of  which 
is  a  function  of  temperature  only. 

It  must  be  clearly  understood  that  the  internal  energy  of 
vapours  and  liquids  (which  physically  depart  far  from  the 
concept  of  a  perfect  gas)  does  depend  upon  volume  as  well  as 
upon  temperature.  From  the  molecular  standpoint  this  is  due 
to  the  existence  of  cohesive  forces  (often  of  very  great  magni- 
tude) between  the  molecules  themselves. 

It  must  be  borne  in  mind,  however,  that  the  given 
definition  of  a  perfect  gas,  though  true  as  far  as  it  goes,  is  not 
a  complete  thermodynamical  definition.  The  complete  defini- 
tion will  be  given  after  we  have  considered  the  porous  plug 
experiment  of  Joule  and  Thomson  (afterwards  Lord  Kelvin). 


CYCLICAL  PROCESSES  OR  CYCLES. 
Reversible  and  Irreversible  Cycles. 

A  cyclic  process  consists  of  a  series  of  consecutive  changes 
or  equilibrium  states  through  which  a  system  may  be  taken, 
finally  returning  to  its  original  position  or  state.  The  system 
is  chemically  and  physically  identical  in  all  respects  at  the 
beginning  and  the  end  of  the  cycle ;  the  system  having 


30         A   SYSTEM  OF  PHYSICAL    CHEMISTRY 

returned  to  its  original  state.  It  is  clear,  therefore,  that  the 
internal  energy  of  the  system  is  the  same  before  and  after  a 
cycle,  for  the  internal  energy  is  a  physical  or  chemical  property 
of  the  substance  just  as  its  volume  or  colour  or  shape  is. 
That  is,  on  completing  a  cycle  we  can  say  that  ±  U  —  o, 
where  ±  U  denotes  the  change  in  internal  or  total  energy,  and 
this  is  true  for  systems  of  any  kind.  The  terms  Q  and  A  may, 
however,  not  be  zero,  for  these  are  not  chemical  characteristics 
or  properties  of  the  system  in  the  same  sense  that  the  internal 
energy  is.  When  a  system  is  undergoing  any  change,  heat 
may  be  added  or  given  out  and  similarly  work  may  be  done 
by  or  upon  the  system. 

The  First  Law  of  Thermodynamics,  when  applied  to  a  cycle^ 
states  that  the  sum  of  all  the  work  terms  (27 A)  is  equal  to  the  sum 
of  all  the  heat  terms  (ZQ). 

This  is  evident,  for  the  First  Law  states  that  for  any  process 
U  =  A  —  Q,  and  since  for  a  completed  cycle  U  =  o  it  follows 
that  A  =  Q  or  27A  —  27Q,  when  during  the  cycle  there  are 
several  operations  involving  work  or  heat.  This  holds  good 
no  matter  what  the  system  may  be  (solid,  liquid,  or  gas,  hetero- 
geneous or  homogenous),  and  it  likewise  holds  good  whether 
the  temperature  of  the  system  has  altered  at  any  stage  of  the 
process  or  not  (the  temperature  at  the  final  stage  being,  of 
course,  equal  to  the  initial  temperature).  A  cycle  in  which  the 
temperature  does  not  change  during  any  period  of  the  opera- 
tion is  called  an  isothermal  cycle.  A  cycle  in  which  tempera- 
ture changes  do  occur  is  called  a  non-isothermal  cycle  (the 
original  conditions  of  temperature  being  eventually  reached). 
A  reversible  cycle  is  one  in  which  the  various  occurring 
processes  are  reversible  in  the  sense  already  defined.  An 
irreversible  cycle  is  one  in  which  the  processes  are  irreversible. 
The  First  Law  of  Thermodynamics  being  an  absolutely  general 
law,  embodied  in  the  principle  of  the  Conservation  of  Energy, 
must  hold  for  reversible  and  irreversible  processes  and  cycles 
alike.  When  we  come  to  the  Second  Law  of  Thermodynamics 
we  shall  see  that  we  must  restrict  ourselves  to  reversible 
cycles.  A  typical  isothermal  cycle  might  be  represented  as 
follows : — 


REVERSIBLE   CYCLES  31 

The  initial  system  is  a  vessel  of  liquid  water  and  from  this 
we  remove  by  isothermal  vaporisation  at  a  given  pressure  one 
gram-mole.  We  now  isolate  the  vapour  and  alter  its  pressure, 
thereby  doing  work  against  the  surroundings  until  its  pressure 
becomes  identical  with  the  vapour  pressure  of  a  given  dilute 
aqueous  solution  contained  in  a  second  vessel  at  the  same 
temperature.  We  now  isothermally  compress  the  gram-mole 
of  vapour  into  the  solution  and  then  by  means  of  a  semi- 
permeable  membrane  remove  one  gram-mole  of  liquid. 
Having  isolated  it,  transfer  it  to  the  liquid  water  contained  in 
the  first  vessel.  The  gram-mole  of  water  has  now  been  taken 
round  a  complete  cycle  (which  happens  in  this  case  to  be  an 
isothermal  one),  the  initial  and  final  states  being  identical.  If 
we  had  carried  out  each  single  process  reversibly  (in  the 
manner  described  in  the  paragraph  dealing  with  the  production 
of  maximum  work  in  a  process  involving  more  than  one  stage, 
and  also  in  the  treatment  of  maximum  osmotic  work  pro- 
production),  the  cycle  as  a  whole  would  be  called  an  isothermal 
reversible  cycle. 

Professor  Orr  (Notes  on  Thermodynamics  for  the  use  of 
Students  in  the  Royal  College  of  Science^  Ireland^  printed 
privately)  defines  reversibility  as  follows  : — • 

"  A  process  is  said  to  be  reversible  in  the  thermodynamic 
sense,  if  it  is  possible  for  the  successive  stages  to  occur  in  exactly 
the  reverse  order  in  point  of  time  (all  the  motions,  chemical 
changes,  transference  of  electricity,  etc.,  being  thus  reversed), 
with  all  the  mechanical  forces  unaltered.  (Care  must  be  taken 
not  to  express  the  definition  in  such  a  way  as  would  imply  a 
reversal  of  the  forces.")  " 

It  must  be  borne  in  mind  that  a  reversible  process  or  cycle 
is  a  limiting  case  which  we  can  never  quite  realise  in  practice. 
All  naturally  occurring  or  spontaneous  processes  are  irreversible. 
We  have  already  defined  an  irreversible  process  as  one  in  which 
energy  is  dissipated  or  departs  from  the  system  permanently 
(though,  of  course,  it  cannot  be  destroyed,  but  must  appear 
somewhere  in  space  according  to  the  principle  of  the  First  Law). 
The  conductance  of  heat  from  a  hot  system  to  cold  without 
the  performance  of  any  external  work  is  an  irreversible  process. 


32         A    SYSTEM   OF  PHYSICAL   CHEMISTRY 

That  is  to  say,  it  is  a  matter  of  experience  that  a  bar  of  metal 
if  heated  at  one  end,  say,  will  not  remain  in  this  state  of 
unequal  temperature,  fc^t  the  temperature  will  ultimately  tend 
to  become  uniform,  the  temperature  of  the  bar  as  a  whole 
being  lower  than  that  of  the  initially  heated  portion.  Further, 
experience  has  shown  that  the  reverse  process  never  takes 
place  spontaneously,  i.e.  a  bar  of  metal  at  uniform  temperature 
never  of  its  own  accord  begins  to  rise  in  temperature  at  one 
end  and  cool  at  the  other.  The  process  of  conductance  is  in 
fact  an  irreversible  one.  Further,  motion  turned  into  heat  by 
friction  is  also  an  irreversible  process.  The  reverse  operation, 
that  of  a  body  set  in  motion  merely  by  heating,  never  occurs. 
Also  the  process  of  diffusion  of  a  gas  or  a  solute  in  solution  from 
a  region  of  high  pressure  or  concentration  to  a  region  of  lower 
pressure  or  concentration  is  an  irreversible  process  as  it  occurs 
in  nature,  i.e.  the  reverse  process  is  unthinkable  as  a  spon- 
taneous effect.  Similarly,  a  gas  expanding  into  a  vacuum  (which 
is  an  extreme  case  of  the  foregoing  process)  is  irreversible.  Of 
course,  it  must  not  be  forgotten  that  processes  such  as  the 
dilution  of  a  solution  or  the  expansion  of  a  gas  can  be  carried 
out  in  a  reversible  manner,  or  rather  we  can  conceive  of  such 
an  operation  as  a  limiting  case  under  certain  given  conditions, 
i.e.  the  conditions  which  must  be  complied  with  to  yield  the 
maximum  amount  of  external  work.  Natural  or  spontaneous 
processes  never  perform  the  maximum  amount  of  work,  and 
may  perform  no  external  work  at  all.  For  our  purpose,  how- 
ever, reversible  processes  and  cycles  are  the  more  important 
as  the  application  of  thermodynamics  to  physical  and  chemical 
problems  concerns  itself  with  these  alone. 

DIFFERENT  FORMS  OF  EXTERNAL  WORK. 

So  far  we  have  considered  the  symbol  A  to  refer  to  the 
work  done  in  the  expansion  of  a  gas  against  an  external 
pressure,  or  the  work  done  in  diluting  a  solution  also  against  a 
pressure.  This  is  mechanical  work.  It  must  not,  however,  be 
concluded  that  such  operations  as  these  are  the  only  conceiv- 
able form  of  external  work.  A  may  also  in  certain  cases  be 


FORMS   OF  EXTERNAL    WORK  33 

measured  by  electrical  work  or  output  of  electrical  energy  if 
the  system  considered  is  capable  of  being  set  up  in  the  form 
of  a  voltaic  cell.  Electrical  energy  is  the  product  of  electro- 
motive force  into  current.  It  is  usual  to  take  as  the  unit  of 
current  the  faraday,  which  is  equivalent  to  96,540  coulombs, 
and  is  the  quantity  of  current  required  to  deposit  one  gram 
equivalent  of  a  metal  or  non-metal,  e.g.  silver  or  iodine.  If  we 
consider  the  chemical  reaction  represented  by  the  stoicheio- 
metric  equation — 

Zn  +  CuSO4aq  =  ZnSO4aq  +  Cu 
which  may  be  written  in  terms  of  ions  as  : 

Zn  +  Cu++  =  Zn+  +  +  Cu. 

(metal)  (metal) 

This  reaction  can  be  made  to  take  place  in  such  a  way  as  to 
yield  electrical  energy.  The  system  is  simply  the  Daniell  cell. 
If  E  is  the  electromotive  force,  then  E  will  also  numerically 
represent  the  electrical  energy  connected  with  the  deposition 
of  one  equivalent  of  copper  in  the  metallic  state,  and  simul- 
taneously the  dissolution  of  one  equivalent  of  zinc  from  the 
metallic  state,  for  the  quantity  of  current  involved  in  this  is 
one  faraday,  and  hence  the  electrical  energy  is  E  X  i  =  E. . 
This  current  can  do  external  work,  such  as  that  of  driving  a 
motor.  When  it  is  made  to  do  maximum  work  the  cell  is 
acting  reversibly,  and  we  are  dealing  with  a  reversible  process 
which  experience  has  shown  is  capable  of  doing  a  considerable 
amount  of  work,  although  the  volume  remains  practically 
constant. 

Another  form  which  A  may  take  is  work  done  against 
gravity.  Thus  let  us  take  the  surface  layer  of  a  liquid  as  our 
system.  This  possesses  free  or  available  capillary  energy 
which  manifests  itself  by  the  phenomena  of  surface  tension. 
The  free  surface  energy  in  this  case  can  be  measured  by 
making  it  do  work  against  gravity  by  drawing  a  column  of 
liquid  up  a  narrow  tube.  If  h  is  the  height  of  the  column 
when  the  meniscus  becomes  steady,  p  the  density,  and  g  the 
acceleration  of  gravity,  the  external  work  done  by  the  surface 
T.P.C.— ii.  D 


34         A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

energy  is  (hp .  g)  work  units.  Free  energy  therefore  can 
manifest  itself  in  all  types  of  external  work.  If  the  tempera- 
ture is  kept  constant  an$  the  work  be  maximum  work,  then,  no 
matter  what  the  nature  of  the  external  work  may  be,  it  gives 
quantitatively  the  decrease  in  free  energy  of  the  system  and 
its  surroundings. 

Before  passing  on  to  the  consideration  of  the  Second  Law 
of  Thermodynamics  it  may  be  well  to  state  once  more  the 
First  Law,  viz. :  IF  OR  WHENEVER  heat  is  converted  into  work  of 
any  kind,  or  work  into  heat,  there  is  always  a  definite  quantitative 
relationship  between  the  heat  which  has  disappeared  as  such  and 
the  work  which  has  been  done,  or  vice  versa. 


THE  SECOND  LAW  OF  THERMODYNAMICS. 

It  will  be  noticed  that  in  the  definition  of  the  First  Law 
given  above,  stress  has  been  laid  on  the  words  "  if  or  when- 
ever" heat  is  converted,  etc.  Experience  has  shown  that 
work  is  always  capable  of  being  transformed  into  heat,  but 
this  is  not  always  the  case  for  the  opposite  transformation,  that 
of  heat  into  work.  Thus  whilst  the  Law  of  the  Conservation 
of  Energy  furnishes  us  with  the  relationships  which  must 
necessarily  hold  when  one  form  of  energy  is  transformed  into 
another,  the  so-called  Second  Law  brings  out  a  peculiarity  which 
belongs  alone  to  energy  in  the  form  of  heat  as  regards  its  con- 
vertibility into  other  forms  of  energy.  The  Second  Law  as  a 
statement  of  the  results  of  experience  teaches  us  that  there  are 
certain  definite  limitations  in  connection  with  the  transforma- 
tion of  heat  into  other  forms  of  energy.  Qualitatively  one  may 
state  it  thus  : 

External  work,  such  as  the  kinetic  energy  of  moving  bodies, 
may  be  transferred  in  many  ways,  and  completely  into  another 
form,  heat  for  example;  but  conversely  the  reconversion  of 
heat  into  work  is  not  a  complete  one,  and  may  not  even  be 
possible  to  the  slightest  extent.  Remember  this  does  not  in 
any  way  contradict  the  First  Law.  For  whether  the  heat  trans- 
formed be  small  or  great  the  First  Law  holds  absolutely,  in 
that  this  small  or  great  heat  change  is  transformed  quanti- 


SECOND   LAW   OF   THERMODYNAMICS          35 

tatively  into  some  equivalent  form  of  energy,  say  motion. 
The  First  Law  simply  states  that  if  or  whenever  heat  is 
transformed  an  equivalent  of  some  other  energy  appears. 
It  does  not  state  that  heat  under  all  conditions  can  be  trans- 
formed. The  distinction  is  most  important.  As  an  example 
of  heat  that  no  one  has  been  able  to  utilise,  i.e.  transform,  we 
might  take  the  heat  energy  of  the  ocean.  We  know  that 
it  is  at  a  certain  temperature,  and  that  it  requires  energy  to  be 
added  to  it  to  raise  its  temperature.  Hence  it  must  contain 
energy  in  the  bound  state.  No  one  has  by  any  device  been 
able  to  use  the  heat  energy,  say  to  drive  the  screw  of  a  vessel 
in  the  ocean  itself.  There  are  numerous  other  instances,  and 
we  have  gradually  come  to  the  conclusion  that  for  some 
reason  or  other  we  cannot  always  convert  heat  into  work. 
This  naturally  leads  to  the  question  :  What  is  the  condition 
which  determines  this  conversion  or  non-conversion  of  heat 
into  work  ?  The  answer  is  simple,  it  is  a  question  of  tempera- 
ture or  rather  temperature  differences.  We  cannot  convert 
heat  into  work  unless  we  allow  this  heat  to  pass  from  a  high 
temperature  to  a  lower  one.  The  efficiency  of  the  process, 
i.e.  the  fraction  of  heat  converted  into  work  is  dependent  on 
the  temperature  difference  between  the  hottest  and  coldest 
part  of  the  system.  This  is  why  kthe  heat  of  the  ocean  is 
unavailable ;  we  ourselves  are  living  at  the  same  temperature 
(or  even  higher).  The  ship,  for  example,  is  at  the  same  tempera- 
ture as  the  ocean,  and  hence,  if  the  above  principle  is  correct, 
one  cannot  expect  the  heat  of  the  ocean  to  be  utilisable  by  the 
ship.  On  the  other  hand,  the  inhabitants  of  a  colder  planet 
might  readily  utilise  our  ocean  heat,  and,  indeed,  the  heat  of 
our  planet  itself,  for  work,  if  only  the  mechanical  difficulties 
could  be  overcome. 
.  The  Second  Law,  as  stated  by  Clausius,  is  as  follows  : — 

"  It  is  impossible  for  a  self-acting  machine  working  in  a 
cycle,  unaided  by  any  external  agency,  to  convey  heat  from  a 
body  at  a  low  temperature  to  one  at  a  higher  temperature ;  or 
heat  cannot  of  itself  (i.e.  without  the  performance  of  work  by 
some  external  agency)  pass  from  a  cold  to  a  less  cold  body." 
First  note  the  phrase  "  of  itself" — heat  cannot  of  itself  pass 


36         A    SYSTEM   OF  PHYSICAL   CHEMISTRY 

from  a  low  to  a  high  temperature.  We  have  seen  that  by 
means  of  an  adiabatlc  compression  of  a  system  the  tempera- 
ture rises,  heat  being  "evidently  thereby  carried  up  the  tem- 
perature scale.  But  this  does  not  contradict  the  above 
statement  about  the  natural  flow  of  heat,  for  we  have,  by  the 
aid  of  external  agency,  had  to  do  work,  namely,  compress  the 
system  in  order  to  make  the  temperature  rise.  It  is  a  known 
experimental  fact  that  systems,  say  gases,  naturally  tend  to 
expand,  the  molecules  tend  to  fly  apart  and  not  to  contract. 
This  fact  is  evidence,  of  a  molecular  kind,  for  the  Second  Law, 
for  by  expanding  the  system  cools.  So  much  for  the  qualita- 
tive statement  of  experience.  We  now  come  to  the  question 
of  a  quantitative  statement.  It  must  be  clearly  understood 
that  the  quantitative  formulation  of  the  Second  Law  (which  will 
be  given  in  a  moment)  holds  only  for  a  reversible  cycle  of 
changes.  We  have  already  discussed  changes  of  this  nature. 
One  other  instance  may  be  given.  Suppose  a  system  of  any 
kind— an  engine,  as  it  is  called — takes  in  heat  Qx  from  the 
boiler  at  temperature  T1  and  drives  a  piston  and  crank,  thereby 
doing  external  work,  and  then  gives  out  heat  Q2  to  the  con- 
denser at  temperature  T2,  the  amount  of  heat  given  out  to 
perform  work  is  Q!  —  Q2  and  this  will  be  quantitatively  the 
work  done ;  for  friction  is  entirely  absent,  as  we  suppose  it  to 
be  an  ideal  engine.  If,  now,  some  external  agency  performs  a 
quantity  of  work  on  this  engine  numerically  equivalent  to 
Qi  —  Q2  and  if  the  engine  thereby  takes  in  heat  Q2  at  T2  and 
gives  up  Qx  at  the  high  temperature  TI}  then  the  engine  is  a 
reversible  one.  When  the  engine  is  working  directly  it  must 
be  doing  maximum  work,  i.e.  just  able  to  overcome  opposing 
forces  in  order  to  comply  with  the  condition  of  working 
reversibly. 

The  efficiency  of  any  engine  is — 

work  done 

heat  taken  in  from  boiler  at  Tj 

It  is  identical  with  Qi  — Q2 

Qi 

Now  there  is  a  theorem  called  Carnot's  Theorem,  the  validity 


SECOND   LAW  OF   THERMODYNAMICS          37 

of  which  depends  on  the  trustworthiness  of  the  Second  Law. 
This  theorem  states  that  a  reversible  engine  has  the  maximum 
efficiency,  and  further,  that  the  efficiency  of  all  reversible 
engines  working  between  the  same  temperature  limits  is  the  same. 
This  holds  good  whatever  the  nature  of  the  substance  doing 
work.1  It  can  be  shown  that  the  efficiency  of  a  reversible 
engine  is  connected  with  the  temperature  limits  referred  to  by 
the  expression — 

Ti  — T2 
*?  =      T~ 

r\     r\  »p    rp 

So  that  we  may  write  ~—^  =    1       ~ 

Qi  AI 

If  we  take  the  case  in  which  the  highest  temperature  of  the 
engine  or  system  is  T  and  the  difference  between  this  tem- 
perature and  the  coldest  temperature  is  </T,  then  the  amount  of 
heat  converted  into  work  is  dQ  instead  of  QJ  —  Q2  and  the 
above  relation  takes  the  form — 


But,  according  to  the  First  Law,  the  heat  dQ  which  has  been 
converted  into  work  may  be  put  identical  with  an  external 
work  quantity  d&  if  there  has  been  no  loss  by  friction,  and  if 
in  fact  dh.  represents  the  maximum  amount  of  work  dynami- 
cally equivalent  to  dQ?  the  above  relation  takes  the  form  — 


for  all  substances  traversing  a  reversible  cycle.  In  words,  the 
term  dh.  represents  the  maximum  amount  of  work  which  can 
be  obtained  from  an  engine  working  between  the  temperature 
limits  T  and  T  —  dT,  the  heat  taken  at  the  high  temperature 
T  being  Q.  This  may  be  taken  as  the  quantitative  statement 
of  the  Second  Law.  If  the  engine  is  not  a  reversible  one  it 
will  in  general  do  less  work  (according  to  Carnot,  it  can 

1  This  is  discussed  in  a  more  advanced  treatment  given  later. 

2  i  calorie  =  4-2  joules  =  4-2  x  io7  ergs. 


38         A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

never  do  more),  and  in  such  a  case  we  can  only  write  the 
inequality  — 


As  before  mentioned,  we  shall  confine  our  attention  to 
reversible  processes  only. 

//"T 
DEDUCTION  OF  THE  EXPRESSION  dh.  =  Q  ---  FOR  THE  CASE 

OF  A  PERFECT  GAS. 

We  have  briefly  indicated  how  the  above  relation  is 
obtained  in  general  for  any  system  undergoing  a  reversible 
cycle  of  changes.  It  will  be  more  easily  grasped,  perhaps,  if 
we  now  obtain  again  the  same  expression  for  the  concrete 
case  of  a  perfect  gas  engine.  Let  us  suppose  the  gas  is 
enclosed  in  a  cylinder  fitted  with  a  weightless,  frictionless, 
piston.  Suppose  we  are  dealing  with  one  mole  whose  volume 
is  vi  at  temperature  T.  Let  the  gas  expand  isothermally  from 
^!  to  the  volume  z'2.  The  maximum  work  which  could  be 
done  by  the  gas  in  such  an  isothermal  change  is  — 


Now  by  the  First  Law— 

U  =  A-Q 

but  we  have  kept  the  temperature  constant,  and  hence,  since 
U  (which  is  change  in  internal  energy)  depends  only  on 
temperature  and  is  independent  of  volume  for  a  perfect  gas 
(as  we  have  already  seen),  it  follows  that,  in  the  present  case  — 

U  =  o  and  therefore  A  =  Q 
U  A  =  Q  =  RTlog^2     ......     (i) 

?~R**S 

Q   (with   a    +    sign)    denotes   the   heat   taken   in   from   the 


SECOND  LAW  OF   THERMODYNAMICS          39 

reservoir  while  the  work  was  being  done  at  constant  tem- 
perature. 

This  important  formula  *  does  not  by  any  means  say  that 
the  work  A  must  always  be  done  when  a  gas  expands  iso- 
thermally  from  the  smaller  volume  v±  to  the  larger  volume  z>2. 
We  have  only  need  to  call  the  Gay-Lussac  or  Joule  experiment 
to  mind  (namely,  the  expansion  of  a  gas  into  a  vacuum)  in 
order  to  see  that  the  term  A  can  be  equal  to  zero.  Every 

possible  value  of  A  between  zero  and  RT  log  —  is  not  only 

vl 

conceivable   but  is  also  attainable.     The  value  RT  log— is, 

however,  a  maximum  one.  If  we  try  to  bring  the  gas  back 
isothermally  from  the  greater  volume  z>2  to  the  initial  volume 
#!  we  need  to  expend  more  or  less  work  depending  upon  the 
efficiency  of  our  compressing  machine.  But,  here  again,  if  the 
efficiency  of  the  engine  (the  gas)  is  a  maximum,  the  minimum 
work  which  the  compressing  machine  has  to  do  to  bring  the 
gas  back  to  its  original  state  is  at  least  A.  The  maximum 
production  of  work  by  the  perfect  gas  engine  is  therefore  that 
which  accompanies  a  process  taking  place  reversibly. 

Now  let  us  consider  the  gas  as  carrying  out  the  same 
operation  at  a  lower  temperature  T  —  dTT,  the  same  mass  of 
gas  working  between  the  same  volume  limits  vl  and  v2  as 
before.  The  maximum  work  done  by  the  system  is  now — 

(A  —  ,/A)  =  R(T  —  </T)  log  ^  per  mole  .     .     (2) 

z/i 

Note  particularly  that  the  volume  limits  are  the  same  even 
though  we  have  considered  the  two  cases  at  different  tem- 
peratures. There  is  no  volume  change  due  to  temperature 
assumed  or  involved  in  the  above  considerations. 

If  we  now  subtract  equation  (2)  from  equation  (i)  we 
obtain — 

dK  =  R</T  log  -2 
*l 

1  Haber,  Thermodynamics  of  Technical  Gas  Reactions,  English  Edition, 
p.  18. 


40         A    SYSTEM   OF  PHYSICAL    CHEMISTRY 


which  is  the  expression  we  have  sought. 

This  expression  can  now  be  slightly  transformed  into  — 


The  differential  on  the  left-hand  side  is  a  partial  one. 
The  expression  stands  for  the  change  in  the  maximum  work 
which  takes  place  when  the  process  is  carried  out,  first  at  one 
temperature  and  then  at  a  temperature  i  deg.  lower.  The  work 
has  been  carried  out  between  the  same  volume  limits  at  each 
temperature,  i.e.  a  change  in  temperature  does  not  involve  a 
change  in  volume  although  it  does  involve,  of  course,  a  change 
in  maximum  work,  for  the  pressure  is  different  in  the  two  cases. 
The  restriction  in  regard  to  working  between  the  same  volume 
limits  in  the  work  process  at  the  different  temperatures,  which 
is  self-evident  in  the  particular  case  of  the  gas,  is  a  general 
restriction,  whatever  the  working  substance  may  be. 

THE  GIBBS-HELMHOLTZ  EQUATION. 
The  First  Law  states  that — 

U  =  A  —  Q 

From  the  Second  Law  one  can  deduce  for  a  reversible 
cycle — 


and  hence  both  combined  for  a  reversible  cycle  give — 

This  is  known  as  the  Gibbs-Helmholtz  equation  and  is   the 


THE   GIBBS-HELMHOLTZ  EQUATION  41 

most  important  thermodynamical  equation  from  the  standpoint 
of  the  application  of  thermodynamics  to  chemical  problems. 
It  gives  the  true  quantitative  relation  between  the  change 
in  internal  energy  U  and  the  change  in  free  energy  A  which 
occur  during  any  given  reversible  process.  If  the  process  is 
of  such  a  nature  that  the  free  energy  does  not  alter  with 

r?A 

temperature,  or  only  very  slightly,  the  term  —  will  be  zero, 

a  A 

or  practically  so.  In  such  a  case  A  =  U,  i.e.  the  free  energy 
becomes  identical  with  the  change  in  internal  energy.  This 
happens  to  be  nearly  true  in  the  case  of  the  chemical  reaction 
occurring  in  the  Daniell  cell,  but  it  must  only  be  regarded  as 
being  of  an  accidental  character.  In  general  A  and  U  are  not 
numerically  identical,  the  quantitative  relation  between  them 
being  given  by  the  Gibbs-Helmholtz  equation. 

The  equation  referred  to  may  be  slightly  modified  as 
follows.  We  have  seen,  in  dealing  with  the  expression  for 
the  First  Law,  namely— 

U  =  A-Q 

that  we  can  also  write  this  in  the  form — 
_Q.  =  A-Q 

where  —  Qv  represents  the  heat  evolved  when  the  process 
occurs  without  alteration  in  volume,  this  being  identical  with 
the  decrease  in  internal  energy  U.  Substituting  —  Qv  for  U 
in  the  Gibbs-Helmholtz  equation,  we  get — 


APPLICATION  OF  THE  EXPRESSION  </A  =  Q  ---  TO  SOME 
REVERSIBLE  CYCLES.     THE  CARNOT  CYCLE. 

First  let  us  consider  an  isothermal  reversible  cycle.  In 
this  case  dTT  =  o,  and  therefore  //A  =  o.  Hence  we  conclude 
that  in  a  completed  reversible  isothermal  cycle  the  sum  of  all 
the  work  terms  done  by  or  on  the  system  cancel  one  another 


42 


A    SYSTEM  OF  PHYSICAL   CHEMISTRY 


and  equate  to  zero.     This  conclusion  is  of  great  importance, 
and  receives  frequent  application. 

Let  us  next  considej  a  particular  type  of  a  non-isothermal 
reversible  cycle  consisting  of  an  isothermal  expansion  of  a 
system  (solid,  liquid,  or  gas),  followed  by  an  adiabatic 
expansion,  this  in  turn  being  followed  by  an  isothermal 
compression,  and  this  by  an  adiabatic  compression,  thereby 
bringing  the  system  back  to  its  original  state.  Such  a  cycle, 
consisting  of  two  isothermal  volume  changes  and  two  adiabatic 
volume  changes,  is  called  a  Carnot  Cycle.  Let  us  suppose, 


1  Isothermal 


^Isothermal 


G  H       F     K 

FIG.  46. 


Volume 


for  the  sake  of  the  mental  picture,  that  the  substance  is  a  gas. 
We  shall  not,  however,  assume  any  gas  laws  in  the  first  instance, 
and  hence  the  general  result  will  be  valid  for  any  system. 
Suppose  the  initial  state  is  represented  on  the  pv  diagram 
(Fig.  46)  by  the  point  A. 

First  Step.  —  Suppose  the  system  expands  through  an 
infinitely  small  volume  dv  isothermally  and  reversibly,  the 
work  done  (i.e.  maximum  work)  is  represented  by  the  area 
ABFG.  During  this  expansion  it  has  taken  in  a  quantity  of 
heat  Q  from  the  reservoir.  This  heat  can  evidently  be 


expressed  also  by  the  term  (  ^~)  dv,   the    partial    differential 


THE   CARNOT  CYCLE  43 

dQ  denoting  the  heat  which  has  to  be  taken  in  to  keep  the 
temperature  constant  while  the  volume  increases  by  unity.1 

Second  Step.  —  The  system  expands  adiabatically  (no  heat 
entering  or  leaving)  and  the  temperature  falls  by  dT.  The 
system  is  now  at  the  point  C.  During  the  second  step  the 
system  does  work  represented  by  area  BCKF. 

Third  Step.  —  The  system,  now  at  the  lower  temperature,  is 
isothermally  and  reversibly  compressed,  the  work  done  upon 
it  being  represented  by  the  area  below  CD,  viz  :  CjD^HK. 
It  gives  out  a  quantity  of  heat  at  this  lower  temperature,  which 
is  a  little  less  than  Q. 

Fourth  Step.  —  The  system  is  further  adiabatically  com- 
pressed, the  temperature  rising  until  the  point  A  is  once  more 
reached.  The  work  done  upon  the  system  in  this  step  is 
represented  by  the  area  ADHG. 

The  cycle  is  now  complete,  and  the  net  work  done  by  the 
system  is  the  area  ABCD.  This  area  is  also  the  product  of 
AE  into  FG.  Now  AE  is  the  increase  in  pressure  experienced 
by  the  system  when  it  is  kept  at  constant  volume,  and  its 
temperature  is  raised  by  dT.  We  can  express  this  analytically 
by  saying  that  — 


Further,  the  line  FG  corresponds  to  a  small  volume  change 
dv,  so  that  the  net  work  done  by  the  system,  namely  dA,  can 
be  expressed  — 


1  The  symbol  -j-  denotes  the  change  of  x  with  y,  other  variables  such 

as  z  simultaneously  changing.  The  symbol  —-  denotes  the  change  of  x 
with  y,  all  the  other  variables  (such  as  z)  being  kept  constant.  The 
expression  -£  is  a  partial  differential.  To  indicate  more  clearly  the 

oy 

variable  (z)  which  is  kept,  constant  during  the  change  of  x  with  y,  the 
partial  differential  can  be  written  (  —  )  • 


44         A   SYSTEM   OF  PHYSICAL   CHEMISTRY 

Now  let  us  apply  the  relation  deduced  from  the  Second 
Law  for  a  reversible  cycle,  viz.  — 


Q  is  the  heat  taken  in  at  the  higher  temperature,  and  we  have 
seen  that  this  is,  in  this  cas 
expression  may  be  written — 


//5O\ 

seen  that  this  is,  in  this  case,  \-~)  dv.     Hence  the  above 

\CJ  V/T 


This  holds  for  any  system,  gaseous,  liquid,  or  solid,  homo- 
geneous or  heterogeneous. 

The  meaning  of  (    ^ }    is  the  heat  which  has  to  be  added 

to  a  system  which  is  increasing  by  i  cubic  centimetre  in  order 
to  keep  the  temperature  constant.  It  is  therefore  the  latent 
heat  of  expansion  of  the  system,  and  may  be  denoted  by  /. 
Hence — 


/_T  ^*- 
l\dT 

Remember,  this  latent  heat  of  expansion  may  refer  to  a 
homogeneous  phase  (completely  gaseous,  completely  liquid, 
or  completely  solid).  It  may  also  refer  to  those  cases  of 
change  of  state  in  which  the  term,  "  latent  heat "  is  more 
familiar. 

THE  CLAPEYRON  EQUATION. 
Let  us  apply  the  relation — 


T 

to  the  case  of  a  change  of  state  from  liquid  to  vapour.  If  A  is 
the  ordinary  latent  heat  per  gram  of  the  substance,  it  is  clear 
that  if  z>i  is  the  specific  volume  of  the  liquid,  i.e.  the  volume  of 


.  THE   CLAPEYRON  EQUATION  45 

i  gram,  and  v%  is  the  specific  volume  of  the  vapour,  then  A 
refers  to  an  increase  in  volume  of  v%  —  v-^.     Hence  the  latent 

heat   per   unit   volume-increase,  viz.  /,   is   given   by  -        — . 
Hence  the  above  expression  becomes — 

A 

T(z>2~ 
^ )    represents  the  change  in  vapour  pressure  of  the  liquid 


(BP\  _         A 

W/       Tz/  — 


per  degree  rise  in  temperature.  Since,  however,  we  know  that 
the  pressure  of  saturated  vapour  is  independent  of  the  volume 
of  the  vapour  so  long  as  any  liquid  is  present,  we  need  not 
retain  the  restriction  of  constant  v  in  this  case,  but  may  simply 
write  — 

d  A 


This  is  known  as  the  Clapeyron  equation.  It  allows 
one  to  calculate,  for  example,  latent  heat  of  vaporisation 
at  a  given  temperature  if  we  know  the  vapour  pressure- 
temperature  curve,  and  the  specific  volumes  of  liquid  and 
vapour  respectively.  Note  that  up  to  this  point  we  have  not 
assumed  that  the  vapour  obeys  the  gas  laws,  or,  indeed,  any 
law.  The  expression  may,  however,  be  made  more  amenable 
to  calculation,  and  may  still  be  regarded  as  accurate  for  all 
ordinary  purposes  if  such  an  assumption  be  now  made. 
Further,  let  us  neglect  the  volume  of  the  liquid  z/j  compared 
to  that  of  the  vapour  v2*  which  is  quite  justifiable  as  long  as 
we  are  at  temperatures  considerably  below  the  critical  point 
(at  which  point  v1  =  z>2)«  According  to  the  gas  equation 

"PT1 

^2  =  -  )  and  hence  the  Clapeyron  equation  becomes  : 

dp  _    \p 


°r 


/VT      RT2 
d  log  p        A 


46         A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

If  we  consider  i  mole  as  the  unit  of  mass,  then  A  will  be 
molecular  heat  of  vaporisation,  and  R  will  be  approximately 
2  calories. 

1.  First  Illustration.  —  At  what  height  must  the  barometer 
stand  in  order  that  water  may  boil  at  101°  C.?     Let  the  rise 
be  x  cms.  of  mercury  for  i°  rise  in  boiling  point  of  water. 
That  is— 

-^  =  x  cms.  =  x  X  13*6  X  981  (dynes  per  degree) 
/  =  i  atmosphere  =  io6  dynes  approximately. 

Hence  i   *  =  '-^a?  X  98. 

/  dT  io6 

Now  A  =  536  calories  per  gram.  R  =  2  calories  per  mole, 
which  equals  ~  calories  per  gram. 

*X  13-6  X  98i_536  X  18 
.*•  —  •  ~-  —  2*7  cms. 

io6  2  X  3732 

Hence  the  barometer  must  fe  787  mms. 

2.  Second  Illustration.  —  What  will  be  the  change  in  freezing 
point  of  water  if  the  pressure  on  the  water  be  increased  by 
i  atmosphere? 

We  can  apply  the  thermodynamic  relation  of  Clapeyron 

dp  A 

in  its  accurate  form,  viz.  -^-=  to  the  phenomenon 

^1  1  (ffj  —  fTj) 

of  fusion. 

The  specific  volume  of  liquid  water  (viz.  z/2)  is  i  c-c.  The 
specific  volume  of  ice  (viz.  z/j)  is  i'i  c.c.  approx.  Hence 
z/2  —  Vi  =  —  o'1  c.c.  (note  the  minus  sign).  A  here  denotes 
the  latent  heat  of  fusion  of  ice,  which  is  80  calories  per  gram, 
or  80  X  4'  2  X  io7  ergs  per  gram. 


Now 

dp 
That  is        80  X  4*2  X  10?  =  273  X  (—  o-i)  X  J 

=  -1-2X108 


THE   CLAPEYRON  EQUATION  47 

The  minus  sign  shows  that  by  putting  on  positive  pressure 
the  temperature  of  fusion  falls^  this  being  due,  as  we  have  seen 
from  the  above,  to  the  fact  that  the  specific  volume  of  water  is 
less  than  the  specific  volume  of  ice.  The  term  i'2  X  io8  repre- 
sents the  pressure  in  dynes — for  all  the  quantities  have  been 
given  in  C.G.S.  units — required  to  lower  the  freezing  point  i°. 

The  reciprocal  of  this,  viz. 


is  the  lowering  of  freezing  point  (in  degrees)  due  to  increasing 
the  pressure  i  dyne.  Since  i  atmosphere  is  io6  dynes,  the 
lowering  of  freezing  point  due  to  an  increase  of  i  atmosphere 
will  be  0*83  X  io~8  X  io6,  or  0*0083°.  So  that  if  the  pressure 
on  freezing  water  be  raised  from  i  atmosphere  to  2  atmospheres, 
the  freezing  point  will  be  —  0*0083°  C.  It  will  be  seen  how 
extremely  small  the  effect  of  pressure  is  on  the  freezing  point, 
i.e.  on  fusion,  this  being  traced  to  the  small  volume  changes 
which  occur  on  fusion ;  on  the  other  hand,  large  volume 
changes  occur  on  vaporisation,  and  therefore  the  effect  of 
pressure  changes  on  the  boiling  point  are,  of  course,  great. 
The  above  considerations  on  the  fusion  point  afford  an  obvious 
explanation  of  the  phenomena  known  as  the  regelation  of  ice. 
In  this  experiment  a  wire  is  hung  over  a  block  of  ice,  the  wire 
being  weighted.  It  is  found  that  the  wire  cuts  its  way  through 
the  ice,  which,  however,  freezes  behind  it  so  that  the  block 
remains  as  a  whole.  The  effect  of  the  wire  is  to  cause  a  local 
increase  in  pressure  on  the  ice.  This  causes  the  ice  to  melt 
in  the  absence  of  an  artificial  lowering  of  the  temperature, 
because  as  we  have  seen  increase  of  pressure  lowers  the  freez- 
ing point,  i.e.  the  equilibrium  temperature  at  which  ice  and 
liquid  co-exist.  The  wire  sinks  in  the  water  thus  formed,  but 
the  super-natant  liquid  being  now  once  more  at  the  original 
pressure,  again  freezes,  since  the  block  as  a  whole  cannot 
be  at  a  higher  temperature  than  o°  C.  It  is  clear,  as  already 
pointed  out,  that  this  phenomenon  of  regelation  is  really 

dp 
dependent  on  the  fact  that  —   is  a  negative  quantity,  and  this 


48         A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

is  dependent  on  the  fact  that  the  specific  volume  of  ice  is 
greater  than  that  of  liquid  water.  In  the  case  of  sulphur,  for 
example,  when  it  reaches  the  temperature  (the  transition  tem- 
perature) at  which  the  rhombic  passes  into  the  monoclinic 
form,  or  the  monoclinic  into  the  liquid,  it  is  found  that  the 
specific  volume  of  the  phase  which  is  more  stable  at  the  lower 
temperature,  is  less  than  the  specific  volume  of  the  phase  which 

is  more  stable  at  higher  temperatures,  and  hence  -—-  is  positive, 

and  so  an  increase  of  pressure  would  raise  the  transition  or 
melting  point.  No  regelation  phenomena  could  possibly 
occur  in  such  a  case. 

From  these  few  illustrations  it  is  evident  that  the  Clapeyron 
equation  is  one  of  the  most  fundamentally  important  thermo- 
dynamic  relations  in  the  study  of  chemical  and  physico- 
chemical  problems.  . 


CHAPTER   II 

FURTHER  CONSIDERATION  OF  THERMODYNAMICAL 
PRINCIPLES. 

ACCORDING  to  the  First  Law,  as  we  have  already  seen,  when- 
ever mechanical  energy  is  converted  into  heat  or  heat  into 
mechanical  energy,  there  is  a  constant  ratio  between  the  two. 
In  a  given  expenditure  of  one  sort  of  energy  we  find  an 
"  equivalent  "  of  some  other  form  of  energy.  Taking  i  calorie 
as  the  unit  of  heat  energy,  it  has  been  shown  that  this  is 
equivalent  to  4*189  joules  'or  4*189  X  io7  ergs  (mechanical 
energy  units). 

Instead  of  making  use  of  the  terms  Q,  U,  and  A  to  denote 
changes  in  the  heat  effect,  internal  energy,  and  external 
work  respectively,  we  shall  simply  use  the  above  terms  to 
denote  heat,  internal  energy,  and  external  work  in  general  ; 
while  to  denote  changes  in  any  of  these  quantities  we  shall 
apply  the  more  mathematically  accurate  form  of  notation,  that 
of  the  differential  calculus.  Thus  the  First  Law  of  Thermo- 
dynamics may  be  stated  in  the  form  of  the  equation — 

^Q  =  </U  +  dk 

which  is  the  same  thing  as  saying,  that  when  one  adds  a  small 
quantity  of  heat  </Q  to  a  system,  there  results  thereby,  a  small 
increase  d\S  in  the  internal  energy  of  the  system,  and  at  the 
same  time  the  system  does  a  small  quantity  dA.  of  external 
work.  Naturally  these  must  be  all  expressed  in  the  same 
units — ergs,  joules,  or  calories — in  order  that  the  two  sides  of 
the  equation  may  be  numerically  identical. 

T.P.C. — ii.  E 


50        A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

THE  FORM  OF  A  OR  dk. 

Any  "  work  term,"  say  </A,  is  always  made  up  of  two  factors, 
as  we  have  seen.  Energy  expended  or  work  done  (by  a  body) 
can  always  be  expressed  as  the  product  of  a  capacity  factor 
into  an  intensity  factor. 

(1)  In  the  case  of  a  very  small  expansion  of  a  system  by 
an  amount  dv  against  a  pressure  /,  the  work — 

/  being  the  intensity  factor,  v  the  capacity  factor. 

(2)  Suppose  a  system,  such  as  a  stretched  string,  is  being 
elongated  by  a  tension  T,  then  if  the  change  in  length  is  dl 
the  work  done  on  the  system  is  T///,  or  the  work  done  by  the 
system  is  — T*//.     In  this  case  T  is  the  intensity  factor,  /  the 
capacity  factor. 

FORM  OF  d\3. 

It  is  also  reasonable  to  assume  that  part  of  ^U  might  be 
regarded  as  expressible  in  terms  of  factors.  But  the  idea  con- 
veyed by  internal  energy  is  a  very  composite  one,  being  more 
composite  in  the  case  of  solids  and  liquids  than  in  the  case  of 
gases,  and  being  likewise  influenced  by  the  molecular  or  atomic 
complexity  of  the  system.  It  is  impossible  to  state  numerically 
in  any  energy  units  what  the  absolute  value  of  U  *  is  for  a 
given  body.  We  can  only  measure  differences  in  U,  namely, 
^U,  in  certain  cases  consequent  upon  changes  in  any  of  the 
factors,  such  as  temperature  or  pressure,  which  define  the  state 
of  a  system. 

VARIABLES  OF  A  SYSTEM. — COMPLETE  AND  INCOMPLETE 
DIFFERENTIALS. 

The  most  familiar  case  in  which  we  have  two  variables,  p 
and  v,  both  depending  on  each  other,  and  also  on  a  third, 

1  Perhaps  it  should  be  again  emphasised  that  the  significance  of  U 
in  this  chapter  and  in  the  succeeding  one,  in  which  we  deal  with  the 
continuity  of  state,  is  not  the  same  as  in  the  preceding  chapter.  The 
"  U  J:  of  the  preceding  chapter  is  here  represented  by  —dU. 


COMPLETE  AND  INCOMPLETE  DIFFERENTIALS  51 


which  we  shall  call  T,  is  the  expression  for  the  behaviour  of  a 
perfect  gas,  namely,  pv  =  RT.  If  we  consider  a  plane  diagram 
in  which  p  is  plotted  against  v,  the  quantity  T  being  kept  con- 
stant, we  obtain  the  familiar  hyperbolic  curve  stretching 
between  any  two  chosen 
points  0,  b  (Fig.  47).  Any 
one  of  the  three  variables  A 
/,  v,  T,  can  be  treated  as 
a  function  of  the  other 
two,  i.e.  T  is  a  function  of 
/  and  v,  or  p  is  a  function 
of  v  and  T,  or  v  is  a 
function  of  /  and  T. 
These  cross  connections 


T  ts  constant 


FIG.  47. 


are  represented  for  the 
particular  case  of  a  per- 
fect gas  by  the  equation 
already  given.  For  a  perfect  gas,  therefore,  if  it  be  taken 
through  a  series  of  changes— expansions,  compressions,  heat- 
ing, cooling — and  again  brought  back  to  its  original  state,  that 
is,  if  we  bring  back  the  pressure  and  volume  to  the  original 
values  pQ  and  £'0,  the  temperature  will  also  have  been  found 
to  have  come  back  to  its  original  value  T0.  In  such  a  case  a 
small  change  in  temperature,  which  is  denoted  by  dt>  is  said  to 
be  a  complete  differential,  because  the  value  of  T  at  any  stage 
is  completely  determined  by  the  values  of  /  and  v  at  that 
stage.  Suppose,  however,  that  in  the  cycle  of  changes  through 
which  we  put  the  gas  there  is  external  work  done,  then  on 
completing  the  cycle  it  will  be  found  in  general  that  the  system 
has  either  done  a  nett  amount  of  work,  or  has  had  a  nett 
amount  of  work  done  upon  it.  If  either  happens,  dK  cannot 
be  regarded  as  a  complete  differential,  because  on  coming 
back  to  its  original  state  the  system  has  suffered  either  a 
permanent  gain  or  loss  of  work.  (In  the  single  case  of  a  per- 
fect gas  going  through  a  cycle  of  operations  at  constant 
temperature  </A  =  o  on  the  whole  cycle.)  The  difference 
between  the  two  cases,  namely,  complete  and  incomplete 
differentials,  as. exemplified  by  dT  and  dK  respectively,  lies  in 


52         A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

this  :  that  the  value  of  </T  depends  only  on  the  position,  say  the 
point  #,  in  the  path  of  transformation.  Whereas  ^A  depends 
on  the  path  itself.  Thijs  if  we  go  from  a  to  b  along  the  path 
indicated  in  the  diagram,  and  then  return  along  the  same  path 
from  b  to  a,  </A  is  certainly  nothing.  But  if  we  return  from  b 
to  a  by  a  different  path — say  one  lower  down  on  the  diagram— 
and  finally  reach  a,  the  total  work  or  JVA  is  not  zero.  J^T  is  zero 
no  matter  what  path  we  follow.  A,  it  must  be  remembered, 
means  external  work  done  by  or  on  the  body.  The  term  U 
denotes  internal  energy  contained  in  the  body.  This  quantity 
U  is  considered  to  be  simply  dependent  on  the  state  of  the 
substance — gas,  liquid,  or  solid — at  any  given  stage.  It  is 
considered  independent  of  the  path  by  which  the  stage  was 
reached.  It  therefore  corresponds  to  the  behaviour  of  T  in 
the  case  of  a  perfect  gas;  ^U  is  therefore  a  complete 
differential.  On  taking  a  body  round  a  cycle  of  changes  and 
bringing  it  back  to  the  original  point,  the  value  of  U  will  once 
more  be  U0,  although,  of  course,  at  different  points  during  the 
cycle  it  possessed  values  sometimes  greater  sometimes  less 
than  this.  One  might  look  then  upon  the  U  of  a  body  as 
something  analogous  to  an  inherent  physical  (or  chemical) 
property  of  the  body.  [The  property  of  boiling  point  is 
always  characteristic  of  a  substance  when  the  pressure  is 
brought  back  to  atmospheric,  no  matter  what  has  happened  to 
the  substance  between  two  boiling  point  determinations,  pro- 
vided, of  course,  it  has  not  been  exposed  to  conditions  so 
extreme  as  to  decompose  it.]  One  must  be  quite  clear  there- 
fore as  to  the  distinction  between  internal  energy  changes  (or 
total  energy  changes,  as  they  are  sometimes  called)  and 
external  work  terms  (or  free  energy  changes  as  they  are  some- 
times called).  The  expression  dU  is  a  complete  differential, 
the  change  in  U  being  completely  determined  by  the  initial 
and  final  states ;  ^A,  on  the  other  hand,  is  an  incomplete 
differential,  the  change  in  A  being  dependent  on  the  path 
whereby  the  transformation  from  the  initial  to  the  final  state 
of  the  system  was  made.  In  any  change  from  the  point  a  to 
point  b,  knowing  the  first  and  last  states,  we  know  d\3,  and 
therefore  we  know  (</Q  —  dA.)}  but  we  do  not  know  either  of 


COMPLETE  DIFFERENTIALS  53 

these  quantities  separately,  unless  one  knows  the  history  of 
the  change.  For  the  given  increase  in  energy  dU  we  cannot 
tell  how  much  of  this  energy  was  given  to  the  body  in  the  form 
of  heat  +  ^Q>  a°d  how  much  was  given  in  the  form  of  work 
done  (—  dh.) l  upon  it  (this  latter  being  transformed  into 
internal  energy). 

Now  let  us  consider  the  heat  term  Q.  As  already  stated 
in  the  preceding  chapter  on  Elementary  Thermodynamics, 
strictly  speaking,  we  cannot  use  the  term  heat  /;/  a  body.  One 
can  pass  heat  into  a  body,  but  it  is  no  longer  heat  when  it  gets 
in.  Take,  by  way  of  illustration,  the  passage  of  heat  into  a  body 
which  is  expanding  at  constant  temperature.  The  heat  has 
simply  gone  to  do  external  work,  and  if  it  is  converted  it  is  no 
longer  "  heat."  Heat  enters  a  system  but  immediately  becomes 
something  which  is  not  heat.  In  general,  there  is  an  increase 
in  the  internal  energy  of  the  system  and  also  some  external 
work  is  done.  In  fact,  we  have  the  relation  dQ  =  d\J  -f  dA. 
Hence,  if  a  piece  of  matter  be  put 'through  a  cycle  of  changes 
and  comes  back  to  its  original  state,  the  internal  energy  U  is 
the  same  as  at  the  beginning,  just  as  the  matter  itself  is  the 
same.  That  is,  the  sum  of  all  the  change  in  U,  namely  ZJdU, 
is  zero  for  the  complete  cycle.  The  sum  of  all  the  work  terms 
done  by  or  on  the  system,  namely  2dA,  is,  however,  not  zero, 
and,  THEREFORE,  from  the  above  equation  dQ  =  dU  +  ^A, 
the  term  dQ  or  27</Q  (if  there  has  been  .more  than  one  heat 
addition  or  subtraction  on  completing  the  cycle),  is  not  zero. 
Hence,  we  cannot  speak  of  heat  in  a  body  in  the  same  sense 
as  we  speak  of  internal  energy  (U)  in  a  body.  Functions  or 
quantities  which  come  back  to  their  initial  values  when  the 
cycle  is  completed  and  the  original  conditions  (say,  of  pressure 
and  temperature)  once  more  obtain  are  called  complete  differ- 
entials•,  and  we  can  express  the  fact  mathematically  in  the  case 
of  a  quantity  x  by  the  equation — 

fdx=o 
for  .a  completed  cycle,  in  which  we  have  once  more  arrived  at 

1  We  reckon  as  a  convention  simply  work  done  by  the  system  as  +, 
work  done  upon  it  as  —  „ 


54        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

the  starting  point.  By  way  of  illustration,  since  we  have 
already  seen  that  internal  energy  change  is  a  complete  differ- 
ential, we  can  write  — 


for  a  completed  cycle.  But  JVA  ^  o  being  an  incomplete 
differential,  and  since  in  the  case  of  a  completed  cycle  JVU  =  o, 
it  follows  that  JWQ  =  /</A,  the  conclusion  being  that  /</Q  =£  o, 
i.e.  </Q  is  an  incomplete  differential. 

PROPERTY  OF  THE  COMPLETE  DIFFERENTIAL. 

If  there  are  two  quantities  x  and  y   upon  which  a  third 
quantity  W  depends,  we  can  express  the  fact  by  the  relation— 

W  =/<*,,) 

Suppose  x  and  y  plotted  as  co-ordinates  in  a  rectangular 
system  (see  Fig.  48). 


FjG.~4S. 

Let  the  initial  state  of  W  be  represented  by  the  point  A 
and  let  its  final  state  in  a  transformation  be  represented  by  the 
point  C.  We  can  pass  from  A  to  C  by  different  paths. 
Consider  two  such  paths,  viz.  one  via  B,  the  other  via  D. 

ist  Case. — Passing  horizontally  from  A  to  B  means  simply 
that  x  has  altered  by  an  amount  dx  while  y  has  remained 
constant.  Passing  from  B  to  C  means  that  y  alters  by  an 
amount  dy  while  x  remains  constant.  The  initial  value  of  W 


COMPLETE   DIFFERENTIALS  55 

at  A  may  be  called  W0,  and  when  it  reaches  B  it  has  evidently 
the  value  — 

(».+£*) 

When  C  is  reached  the  value  of  W  is  — 


ind  Case.  —  Passing  vertically  from  A  to  D  means  that  y 
has  altered  by  an  amount  dy  while  x  has  remained  constant. 
Passing  from  D  to  C  means  that  y  remains  constant  while  v 
changes  by  the  amount  dx.  The  initial  value  of  W  at  A  is  as 

before  W0.     On  reaching  D  it  has  the  value  (  W0  +  y~ 
When  C  is  reached  W  has  the  value  — 


If  W  depends  only  on  the  two  variables  x  and  y,  that  is,  if  W 
is  completely  determined  by  the  instantaneous  values  which 
these  two  variables  possess  at  any  moment,  it  is  evident  that 
on  going  round  any  cycle  on  the  x,  y  diagram  and  returning  to 
A,  fdW  will  be  zero,  i.e.  dW  is  a  complete  differential.  Hence, 
it  does  not  matter  what  path  is  followed  in  going  from  A  to  C, 
the  value  of  W  at  C  will  be  the  same  in  each  case.  That  is, 
expression  (i)  is  equal  to  expression  (2),  or, 


dx\dy 

Remember  this  relationship,  which  we  shall  apply  in  many 
instances  later,  only  holds  good  if  dW  is  a  complete  differential. 
If  this  is  not  the  case,  the  equality  between  expressions  (  i  )  and 
(2)  will  not  necessarily  be  true,  the  discrepancy  between  them 
representing  the  difference  in  the  W  value  caused  by  the 
difference  in  path. 

Illustration  of  the  above  relation.  —  For  a  cycle  completed  by 
any  system,  solid,  liquid,  or  gas,  we  have  seen  that  jdl  =  o, 
i.e.  dl  is  a  complete  differential.  Let  us  take  the  particular 


$6         A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

case  of  a  perfect  gas.  T  being  here  a  function  of  p  and  v 
only,  according  to  the  relation— 

'       T—  &L 
~R 

dT  is  now  the  complete  differential,  and  we  can  in  this  case 
write  T  in  place  of  W,  v  for  x  and  p  for  y  in  the  above 
expression,  whence  we  find  — 

d  /dW\  .  d  (d'Y 

—  (  -r-  )  becomes      (  — 


and  since  T  =—',  it  follows  that  Cr 

K.  op        K 

d(d\V\       d(v\       i 
hence  I  —  )  =  —I  -  )  = 

dx\  d    J      dp\R/      R 


dx\  dy 

Again,  we  also  find  that  — 
d/dW 


. 

becomes    -  1 
dp\ 

and  since  T  =  TV  >  it  follows  that  _-    =^ 

a/<5VV\      B(p\       i 

and  therefore        r-  (  -—  )  =  _  (  ^  -  )  =  - 

d)\dx.'      d/\R/      R 

We  have  therefore  reached  the  same  result  in  both  cases, 
showing  thereby  the  validity  of  the  above  mathematical  relation 
in  this  particular  instance. 


FURTHER  CONSIDERATION  OF  THE  FIRST  LAW  AND  THE 
METHODS  OF  EXPRESSING  IT  IN  DIFFERENT  CASES. 

When  some  heat  is  added  to  a  body  two  changes  will  in 
general  take  place. 

(1)  The  temperature  of  the  body  will  increase. 

(2)  The  volume  of  the  body  will  increase. 

Let  us  regard  these  two  effects,  not  as  occurring  simul- 
taneously but  as  consecutively.  That  is,  we  consider  that  the 
total  effect  produced  is  divided  up  into  (i)  an  increase  of 
temperature,  the  volume  of  the  body  or  system  remaining 


FIRST  LAW   OF   THERMODYNAMICS  57 

constant,  followed  by  (2)  an  hicrease  in  volume,  the  tempera- 
ture remaining  constant.  In  the  first  stage  of  the  process  if 
the  temperature  of  the  body  rises  dt  degrees,  the  heat  which 
has  to  be  added  is  Cy<#>  C«  being  the  specific  heat  at  constant 
volume.  In  the  second  stage  (in  reality,  simultaneously  with 
the  temperature  change),  the  volume  increases  by  an  amount 
dv  at  constant  temperature.  Suppose  /is  the  latent  heat  of 
expansion  of  the  body,  that  is  the  heat  which  must  be  added 
to  the  body  to  cause  it  to  increase  by  unit  volume  while  the 
temperature  is  kept  constant,  then  the  heat  which  is  required 
to  keep  the  temperature  constant  while  the  volume  increases 
by  dv  is  Idv.  Suppose  the  total  amount  of  heat  added  to 
the  system  is  ^Q,  then  it  follows  from  the  principle  involved 
in  the  First  Law,  that  — 


This  relation  holds  good  for  any  system  whatsoever,  gaseous, 
liquid,  or  solid,  and  holds  equally  well  for  homogeneous  and 
heterogeneous  systems.  With  regard  to  the  term  /,  "the 
latent  heat  of  expansion,"  it  is  so  natural  to  connect  latent 
heat  with  a  fusion  or  vaporisation  process  that  the  student 
might  perhaps  think  that  a  heterogeneous  system  only  was 
being  referred  to.  This  is  not  the  case,  as,  for  example,  we 
can  imagine  a  gas  expanding  at  constant  temperature,  heat 
having  to  be  added  in  order  to  keep  the  temperature  constant, 
Now  let  us  link  this  expression  up  with  the  equation  which 
we  have  taken  as  a  statement  of  the  First  Law  of  Thermo- 
dynamics, namely  — 


Since  we  are  considering  a  system  (of  any  sort  whatsoever) 
which  undergoes  a  volume  change  against  a  pressure  which 
may  be  represented  by  /,  we  can  then  write  the  term  pdv 
instead  of  a?A.  The  First  Law  expression  may  therefore  be  put 
into  the  shape  — 

<AJ  =  </Q  —  pdv 

and  substituting  the  value  we  have  found  for  <^Q  in  the  case  of 


58         A    SYSTEM   OF  PHYSICAL    CHEMISTRY 

a  system  which  undergoes  a  temperature  and  volume  increase, 
we  obtain  for  the  increase  in  internal  energy  the  expression— 


Remember  this  expression  holds  for  any  system  or  substance 
whatsoever. 

CRITERION  OF  A  PERFECT  GAS. — THE  JOULE  EXPERIMENT. 

A  perfect  gas,  we  have  seen  from  the  kinetic  standpoint, 
is  one  in  which  molecular  attractions  are  entirely  absent. 
Since  this  is  so,  it  follows  that  no  internal  work  can  be  done 
in  separating  the  molecules  from  one  another  when  the  gas 
expands.  This  means  that  during  the  expansion  no  heat  goes 
to  do  internal  work  ;  the  heat  which  in  the  general  case  has  to 
be  added  to  keep  the  temperature  constant,  namely,  the  latent 
heat  /,  is  now  required  simply  to  enable  the  gas  to  overcome 
the  external  resistance  pressure  p  and  to  do  the  external 
work  pdv  involved  in  the  volume  increase.  That  is  to  say,  we 
can  regard  Idv  and  pdv  as  identical  numerically,  and  therefore 
for  a  perfect  gas  we  can  write  the  equation— 

<tQ  =  Cvdt  +pdv 

This  equation  does  not  hold  for  any  other  system, 
however,  owing  to  the  presence  of  cohesion  or  molecular 
attraction  forces,  say  in  the  case  of  liquids,  solids,  vapours 
and  indeed  ordinary  gases,  though  only  to  a  slight  extent  in 
the  last  named.  We  have  seen  that  for  any  system  whatsoever 
the  increase  in  internal  energy  </U,  when  the  volume  and 
the  temperature  of  the  system  increase,  is  given  by  the 
expression — 

(l  —  p}dv 


Now  we  have  further  seen  that  in  the  case  of  a  perfect  gas, 
there  being  no  internal  work  done  in  expanding  (though  there 
is  internal  energy  represented  by  U),  the  latent  heat  of 
expansion  at  constant  temperature  Idv  for  the  small  volume 
change  is  identical  with  the  external  work  pdv,  at  constant 


THE  PERFECT  GAS  59 

temperature.      Hence  for  a   perfect   gas  the   above  equation 
reduces  to  — 


This  shows  that  the  internal  energy  U  of  a  given  mass  of  a 
perfect  gas  is  independent  of  the  volume  occupied  by  the  gas 
(there  is  no  V  term  in  the  above  equation)  ;  and,  further,  that 
U  depends  only  on  the  absolute  temperature.  When  the 
temperature  of  a  perfect  gas  is  kept  constant,  i.e.  when  dt  =  o, 
then  d\3  =  o,  or  U  remains  constant,  and  this  even  during  an 
expansion  against  external  pressure  /,  when  work  pdv  is  done 
at  the  expense  of  the  latent  heat  which  is  allowed  to  stream  in. 
The  above  criterion  of  a  perfect  gas,  namely,  that  its  internal 
energy  is  independent  of  the  volume,  and  depends  only  on  the 
temperature,  was  first  recognised  by  Joule,  who  carried  out  the 
experiment  described  in  the  preceding  chapter.  The  same 
question  is  taken  up  later  on  (page  68). 

ALTERNATIVE  EXPRESSION  FOR  THE  FIRST  LAW  IN  THE 
CASE  OF  A  PERFECT  GAS. 

It  has  been  pointed  out  that  in  the  case  of  a  perfect  gas 
only,  the  First  Law  can  be  written  in  the  form  <^Q  =  Cydt  -\-pdv. 
Now  we  can  write  this  in  an  alternative  form  involving  Cp 
instead  of  Cv,  where  Cp  stands  for  the  specific  heat  of  the  gas 
at  constant  pressure.  Thus,  since  we  are  dealing  with  i 
gram  of  a  perfect  gas,  then  at  any  stage  in  the  transfor- 
mation considered  we  always  have  the  relation  pv  =  RT,  or 
pdv  +  vdp  =  R<#. 

Further     Cp  —  Cv  =  R  1 

pdv  =  Cpdt  —  Cvdt  —  vdp 

Hence     </Q  ==  Cvdt  -\-pdv  =  Cvdt  +  Cpdt  —  Cvdt  —  vdp 
or  </Q  =  Cpdt  —  vdp 


1  This   may  be  easily  shown   as  follows  :  —  Consider    I  gram  of  gas 
volume  v,  pressure  /,  temperature  T.     We  have  the  relation 

/*  =  RT  .........     (i) 

(R  being   given  the  correct    numerical  value    for  I    gram,  i.e.  R  =  1-9 
calories  per  gram-mole).     Suppose  the  temperature  is  raised  i°  at  constant 


60         A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

Remember  these  relations  hold  only  for  a  perfect  gas.     For 
any  other  system  we  are  only  justified  in  writing  — 


THE  SECOND  LAW  OF  THERMODYNAMICS. 

To   show  that  ^Q  is  not  a   Complete  Differential,  but  that  ~= 

or  d<$>  is  a  Complete  Differential. 

If  we  take  any  system  round  a  complete  cycle,  we  have 
already  seen  that  the  internal  energy  U  returns  to  its  initial 
value,  and  fd\J  =  o,  or  in  other  words  d\3  is  a  complete 
differential.  We  have  seen  likewise  that,  having  completed  a 
cycle,  the  expression  JV/A  is  not  necessarily  zero,  and  since  we 
must  have  by  the  First  Law  of  Thermodynamics  the  expression 
</Q  =  d  U  -f-  ^A,  it  follows  that  JV/Q  may  not  be  zero,  that  is  </Q 
is  not  a  complete  differential.  Instead  of  going  round  a  com- 
plete cycle,  suppose  we  take  the  substance  from  an  initial  to  a 
final  state,  we  see  as  a  consequence  of  the  above  reasoning 
that  the  U  of  the  system  depends  upon  the  state  in  which  it  is 
at  the  moment,  and  that  therefore  the  difference  in  U  due  to 
the  transformation  is  entirely  determined  by  the  initial  and 
final  states  of  the  transformation.  Such  is  not  the  case  with 
the  work  term.  This  is  not  simply  dependent  on  the  initial 
and  final  states  —  say  the  volume  occupied  by  the  system  at 
start  and  finish  —  but  likewise  depends  on  the  path  followed. 
If,  for  instance,  the  temperature  was  kept  constant,  a  change 
in  A  would  have  had  a  certain  value,  but  this  would  be 
different  if  there  had  been  temperature  changes  en  route,  even 
if  the  initial  and  final  volume  had  been  v±  and  z>2  in  both  cases. 

pressure  /,  the  volume  increases  to  vl  and  the  heat  added  is  Cp,  Again 
we  have  the  relation 

M  =  R(T+i)    .......    (2) 

Subtracting  (2)  from  (i)  we  obtain  p(v±  —  v)  =  R.  That  is,  the  work  done 
in  the  expansion  =  R.  But  the  difference  between  the  specific  heat  at 
constant  pressure  and  at  constant  volume  respectively  is  the  equivalent  of 
the  work  done  in  the  expansion  for  i°. 

.*.   C^p  —  Cji  =  K. 


SECOND  LAW  OF   THERMODYNAMICS          61 

Since  A  is  dependent  on  the  path,  it  follows  that  heat  effects, 
positive  or  negative,  are  also  dependent  on  the  path.  Neither 
work  nor  heat  can  in  fact  be  looked  upon  as  intrinsic  properties 
of  the  substance  in  the  same  way  as  internal  energy  can  be 
regarded.  Let  us  consider  a  special  case,  namely,  a  perfect 
gas  undergoing  a  reversible  transformation  in  which  it  does 
maximum  work  in  expanding,  and  let  us  apply  the  First  Law 
in  the  form  here  applicable,  namely  — 


which  may  be   changed   for  a   process   involving   maximum 
work  to  — 

^Q  =  CJt+  *^  =  Cvdt+  RTW  log  v 

Then  considering  the  whole  change  from  an  initial  state  in 
which  the  volume  is  ?'Q  and  the  temperature  T0,  to  a  final  state 
in  which  the  volume  is  v  and  the  temperature  T,  we  can 
write  — 

JVQ  =  fCvdt  +  (  VRTV  log  v 
J  T0  J  VQ 

It  will  be  observed  that  the  numerical  value  of  /C«x#  is 
simply  determined  by  the  limits  of  the  integral  T0  and  T.  In 
the  case  of  the  expression  /RTV/  log  v  the  numerical  value  is 
not  only  dependent  on  the  initial  and  final  values  of  the 
volume,  namely,  V0  and  V,  but  likewise  requires  a  knowledge 
of  all  the  temperature  changes  en  route  ^  for  each  little  d  log  v 
has  to  be  multiplied  by  the  temperature  T  at  that  moment 
before  integration  is  possible.  Hence,  to  evaluate  this  we 
must  know  the  path  as  well  as  the  limiting  states.  This  illus- 
tration makes  it  clear  how  d&  and  therefore  </Q  are  incomplete 
differentials,  and  at  the  same  time  it  suggests  a  change  which 
will  give  us  an  expression  containing  </Q,  but  at  the  same  time 
the  expression  itself  will  be  a  complete  differential.  It  is  seen 

that  RT</  log  v  or  RT  —  can  be  made  a  perfect  differential 

v 

in  certain  cases,  e.g.  \ 

(i)  When  an  isothermal  change  is  considered,  the  tempera- 
ture being  constant  throughout  ;  and 


62          A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

(2)  more  generally,  even  when  the  change  is  non-isothermal, 
if  we  divide  by  T.  The  First  Law  then  takes  the  form  for  a 
perfect  gas  — 


_r. 
"T"     *~TH  ~¥~ 

The  last  term  R  —  no  longer  contains  any  indefinite  factor, 

T   being   now  removed.     Its   integral  can   be  evaluated   on 
simply  knowing  the  initial  and  final  states  as  regards  volume. 

jf-r-t 

Similarly  the  integral  of  Cv.—  can  still,  as  in  the  first  instance, 

be  evaluated  on  simply  knowing  the  initial  and  final  tempera- 
ture states  of  the  system  which  has  undergone  transformation. 

Hence  ~  is  a  complete  differential  depending  upon  the  initial 
and  final  states  only.     In  a  complete  cycle    I  ~  =  o.      The 

quantity  —    is  called  the  change  in  the  entropy  of  the  system, 

and  is  denoted  by  d$.     For  any  change  therefore  from  state 
(i)  to  state  (2) 


For  a  complete  cycle  when  T  and  V  have  come  back  to 

T      V 
their  original  values   the  terms   —  >  become   necessarily 

Ao      vo 

unity  and  their  logarithms  are  zero,  so  that  JV</>  =  o  for  the 
complete  cycle.  Remember  we  have  only  been  considering 
a  reversible  cycle,  and  further,  we  have  restricted  ourselves  to 
a  perfect  gas.  We  shall  see  how  this  latter  restriction  may  be 
justifiably  removed. 

FURTHER  REMARKS  ON  THE  CARNOT  CYCLE. 

For  any  completed  reversible  cycle,  and  therefore  for  the 
particular  case  of  a  Carnot  Cycle  (which  has  been  studied  in 


THE   CARNOT  CYCLE  63 

the  elementary  treatment,  Chap.  I.),  we  know  from  previous 
consideration  that  JVU  =  o,  i.e.  the  U  is  once  more  at  its 
original  value.  Similarly  jd<f>  =  o,  i.e.  the  entropy  of  the  system 
is  once  more  at  its  original  value  when  the  cycle  is  complete. 
Since  internal  energy  and  entropy  depend  only  on  the  initial 
and  final  states,  and  these  states  are,  of  course,  identical  for  a 
complete  cycle,  the  entropy  and  internal  energy  do  not  depend 
on  the  path  followed.  The  expression  jdA.  is,  however,  not 
zero,  i.e.  there  has  been  a  nett  gain  or  loss  of  external  work  by 
the  system,  and  hence  JVQ  is  not  zero ;  there  has  been  a  nett 
addition  or  subtraction  of  heat  energy  to  or  from  the  system 
to  balance  the  work  done  by  or  done  on  the  system  at  some 
stage  or  stages  of  the  transformation.  Let  us  see  what  these 
work  and  heat  terms  are  in  the  special  case  of  a  Carnot  Cycle. 
As  we  passed  isothermally  along  AB  (see  Fig.  46,  p.  42), 
the  system  took  in  an  amount  of  heat  Qx  from  the  infinite  heat 
reservoir.  The  change  in  entropy  in  going  from  A  to  B  is 
therefore 


which,  since  Tj  is  kept  constant,  may  be  written — 
/Q,     or  simply  -^ 


In  passing  along  BC  no  heat  is  taken  in  or  given  out,  and 
therefore  */Q  =  o,  and  hence  the  entropy  change  along  BC  is 
zero,  i.e.  the  entropy  at  C  is  the  same  as  at  B.  (Remember  we 
do  not  know  how  much  this  is,  we  can  only  deal  with  changes  in 

entropy,  such  as  —,-  for  the  AB  transformation.)     In  passing 
Ai 

from  C  to  D  isothermally,  the  entropy  change  is  —  *p 2 ,  the 

2 

temperature  being  maintained  constant,  and  the  minus  sign 

being  used  to  denote  that  the  transfer  of  heat  (in  this  case  an 
evolution  of  heat  by  the  system),  is  in  the  opposite  direction 
to  the  heat  transfer  at  T1}  which  involved  an  absorption  of 


64          A    SYSTEM   OF  PHYSICAL    CHEMISTRY 

heat.    Passing  from  D  to  A  the  heat  transfer  in  either  direction 
is  nil.     On  the  whole  cycle  the  total  change  in  entropy  is— 


and  this  must  be  zero,  since  jd<j>  =  o. 
Hence  i  =  Q? 


Although  the  entropy  change  is  zero  on  the  whole  cycle, 
the  heat  change  is  not  zero,  it  being  evidently  (Qi  —  Q2), 
this  being  the  amount  of  heat  equivalent  to  the  work  done  by 
the  system,  for  fd\J  =  o,  and  always  JVQ  =  JVU  +  /</A.  Cal 
this  nett  total  work  A.  Then 

A  =  Qi  — Q2 (2) 

If  we  suppose  Qj  >  Q2,  Qj  —  Q2  is  positive,  and  nett  work 
is  done  by  the  system.  Substituting  in  (2)  the  value  of  Q2  given 
by  equation  (i),  we  obtain 


A~=  ~\        'lj 

In  words  this  expression  states  that  only  a  fraction  of  the 
heat  Q!  which  is  admitted  to  the  gas  at  the  high  temperature 
Tj  is  converted  into  useful  work.  If  we  express  the  work  as 
a  fraction  of  the  heat  taken  in  at  the  higher  temperature,  we 
have — 

_     _         1          -^2  /    \ 

TV  ~  ~7p  (4) 

This  "  fractional  "  work  is  simply  the  efficiency  rj  of  the 
engine,  and  the  efficiency  of  this  reversible  engine  (i.e.  the  gas) 
depends  therefore  only  on  the  temperature  limits  Tx  and  T2. 
It  does  not  depend  on  the  absolute  size  of  the  engine,  i.e.  on  the 
absolute  amounts  of  heat  taken  in  or  given  out.  The  efficiency 
of  every  reversible  engine  is  therefore  the  same  as  long  as  it 
works  between  the  same  temperature  limits,  i.e*  temperature  of 
the  "  boiler,"  and  the  temperature  of  the  "  condenser."  Besides 
showing  that  the  efficiency  of  every  reversible  engine  was  the 


CARNOrS    THEOREM  65 

same,  if  the  temperature  limits  were  the  same,  Carnot  showed 
that  no  engine  can  be  more  efficient  than  a  reversible  one 
(Carnot's  Theorem).1  Granting  the  validity  of  these  two 
generalisations,  we  see  that  we  need  no  longer  restrict  a  Carnot 
Cycle — or  the  conclusions  which  we  have  come  to  regarding  the 
relation  of  A  to  Qj  —  simply  to  a  perfect  gas.  Any  system, 
solid,  liquid,  or  gas,  may  be  conceived  of  as  being  taken  round 
the  cycle.  Hence  for  any  system  working  as  a  reversible  cycle 
we  have  the  relation — 


or  if  the  temperature  differences  be  denoted  by  dT,  and  the 
work  by  </A,  we  have  — 


Remember  that  A  or  */A,  i.e.  the  area  ABCD,  only  represents 
the  external  or  "useful"  work.  In  the  case  of  a  perfect  gas, 
we  have  no  other  kind  of  work  to  deal  with.  For  an  imperfect 
gas,  or  a  liquid,  etc.,  expansions  and  compressions  involve 
internal  work  as  well.  These,  however,  do  not  enter  into  the 
discussion,  and  do  not  vitiate  the  generality  of  the  results  of 
equations  (3)  and  (4)  (p.  64).  As  long  as  the  process  is  re- 

versible fd(/>  or   I  -~j=  is  zero  for  any  substance,  that  is,  d<j>  is  a 

complete  differential.  Not  only  is  T  an  "  integrating  factor  " 
for  the  equation  of  a  perfect  gas,  it  is  likewise  one  for  all  sub- 
stances, as  follows  from  the  two  generalisations  of  Carnot. 

The  expression  connecting  the  maximum  work  done  by  an 
engine  with  the  heat  taken  in  from  the  boiler  (Qx  at  Tj)  may 
be  regarded  as  a  quantitative  statement  of  the  Second  Law  of 
Thermodynamics,  which  we  have  already  seen  is  stated  in 
general  terms  by  Clausius  thus  :  "It  is  impossible  for  a  self- 
acting  machine  working  in  a  cycle,  unaided  by  any  external 
agency,  to  convey  heat  from  a  body  at  a  low  temperature  to  one 
at  a  higher  temperature,  or  heat  cannot  of  itself  (i.e.  without 

1  For  consideration  of  this  see  Partington's  Thermodynamics. 
T.C.P.  —  II.  F 


66         A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

the  performance  of  work  by  some  external  agency)  pass  from 
a  cold  to  a  warmer  body." 


COMBINATION  OF  THE  FIRST  AND  SECOND  LAWS. 

I.  An  Expression  for  tJie  Latent  Heat  of  Expansion. 

For  any  system,  solid,  liquid,  or  gas,  which  is  undergoing 
a  volume  and  temperature  change,  whether  reversibly  or  irre- 
versibly, we  have  seen  (p.  57)  by  the  First  Law  of  Thermo- 
dynamics that 

d?Q  =  Ctdt  +  Idv 

and  by  a  purely  algebraical  change  we  can  write  this  — 

dQf_d4t     Idv 
T  :  :    T         T 

If  now  we  consider  any  system,  solid,  liquid,  or  gas,  going 
through  a  transformation  with  the  restriction  that  this  transfor- 
mation is  a  reversible  one,  the  Second  Law  of  Thermodynamics 

tells  us  that  the  quantity    *j~   is   a  complete  differential,  and 

we  have  denoted  this  by  the  term  d</>t  which  stands  for  change 
of  entropy  of  the  system.  So  that  both  laws  applied  simul- 
taneously lead  to  the  expression  for  any  system  changing  in  a 
reversible  manner  — 


Since  d§  is  a  complete  differential  we  can  perform  the 
mathematical  operation  already  discussed  in  connection  with 
complete  differentials,  arriving  at  the  expression— 


_        _ 

T  dv~~TdT      T2 

dCv      dl       I 


LATENT  HEAT  OF  EXPANSION  67 

We  have  also  seen  (p.  58)  that  for  any  system — 


(this  being  derived  without  introducing  the  Second  Law). 

Since  dU  is  a  complete  differential,  for  a  complete  reversible 
cycle  it  follows  by  partial  differentiation  that — 

oC/7)  91  dfl 

. == •*  -  (~\ 

Bv        dT      dT  *  ' 

By  combining  expressions  (i)  and  (2)  we  obtain — 

(3) 


This  is  an  important  relation  already  obtained  in  the  pre- 
ceding chapter.  /,  it  will  be  remembered,  is  the  latent  heat 
of  expansion,  namely,  the  heat  required  to  keep  the  tempera- 
ture of  the  system  (solid,  liquid,  or  gas)  constant  while  unit 
increase  in  volume  takes  place.  Note  that  this  is  not  neces- 
sarily latent  heat  of  vaporisation  (i.e.  volume  change  in  a 
heterogeneous  system  containing  liquid  and  vapour),  but 
equally  applies  to  a  volume  change  taking  place  in  a  purely 
homogeneous  system. 

1  The  expression  I  -~-  I    represents  the  change  in  the  specific  heat  at 


constant  volume  with  the  volume,  the  temperature  being  maintained  con- 
stant. The  meaning  of  this  sometimes  is  a  source  of  difficulty.  Suppose 
you  take  a  system  at  (large)  volume  vlt  and  measure  the  specific  heat, 
keeping  the  volume  constant  at  vlt  you  get  a  certain  value  for  C».  Sup- 
pose you  take  the  same  system  at  a  different  volume  vz  (brought  about  by 
a  change  in  pressure)  and  again  measure  the  specific  heat,  keeping  the 
volume  constant  at  v 2,  you  again  get  a  value  for  Cp,  which  may  not  be  the 
same  as  in  the  first  case.  The  difference  of  the  two  values  of  CD  divided 

by  the  difference  of  v±  —  v2  would  be  ~j-  •     If  you  have  carried  out  the  two 

determinations  at  the  same  mean  temperature,  the  difference  of  the  two 

/dCA 

values  of  Cu  divided  by  &i  —  z>2  gives  you  I  "^     I 


68         A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

II.   The  Joule  Experiment  and  the  Criterion  of  a  Perfect  Gas 
from  the  standpoint  o£  the  two  Laws  of  Thermodynamics. 

We  have  already  seen  that  the   increase  in  the   internal 
energy  of  any  system  may  be  expressed  — 


—  p)dv 
Substituting  for  /  its  value  T(  —,  J  we  obtain  — 


Now  it  has  been  shown  that  the  internal  energy  of  a  per- 
fect gas  would  be  independent  of  the  volume  occupied,  and 
would  depend  only  on  the  temperature  (Joule's  Law).  Hence, 
applying  the  above  expression  to  the  case  of  a  perfect  gas,  in 
order  to  make  d\3  independent  of  v>  it  is  necessary  to 
consider  — 

..*•>-• 

This  differential  equation  is  satisfied  by  the  relation— 


where  f(v)  is  simply  an  integration  constant  independent  of 
T.  It  should  be  noted  that  whilst  this  expression  is  a  criterion 
of  a  perfect  gas,  other  substances  might  conceivably  satisfy  it. 
We  shall  come  to  this  point  later  in  discussing  the  Joule- 
Thomson  experiment. 

III.  The  Clapeyron  Equation. 

The  deduction  of  this  expression  need  not  be  further  given, 
as  it  has  already  been  considered  in  the  Elementary  Treat- 
ment. It  is  of  interest,  however,  to  discuss  briefly  the 
quantities  denoted  by  the  terms  internal  and  external  latent 
heat. 


THE    CLAPEYRON  EQUATION 


69 


External  and  Internal  Latent  Heat  of  Vaporisation. 
In  the    process  of  vaporisation,  the  heat  which  we  add, 
namely,  A  calories  per  gram,  to  keep  the  temperature  constant 
may  be  divided  into  two  parts. 

1.  Part  of  the  heat  goes  to  overcome  the  internal  cohesive 
forces  (which  are  very  large  in  the  liquid  state). 

2.  Part   of  the  heat   goes   to   do   the   external   work   of 
expansion  against  the  pressure  exerted  by  the  vapour. 

The  first  part  is  called  the  internal  latent  heat,  and  may  be 
represented  by  A;,     The  second  part  is  the  external  latent  heat 
or  heat  used  purely  in  doing  external  work  (namely,  vapour- 
pressure  X  volume  increase),  and  may  be  represented  by  \ex. 
That  is  A  =  Ai  +  Aeic 

We  can  obtain  an  approximate  value  for  the  external  latent 
heat  as  follows  : — 

Take  the  case  of  water.  Neglect  the  specific  volume  of 
liquid  compared  to  the  specific  volume  of  steam.  Work  of 
expansion  =p(v1  —  v0)=#Vi.  Apply  the  gas  law  to  the 
steam  as  an  approximation.  Then  pv±  =  RT,  where  R  refers 
to  i  gram,  /.<?.  R  =  T2§-  —  J  calories  approx.  Suppose  we  con- 
sider the  boiling  point  of  water.  T  =  373.  Then, 

pv±  —  ^P  =  41  calories  approx.  =  \,x 

The  total  latent  heat  A  is  approx.  536  calories  at  100°  C.,  so 
that  practically  500  calories  (i.e.  Ai)  are  required  for  the  internal 
work  against  the  cohesive  forces.  The  following  table  contains 
some  of  the  values  given  by  Zeuner  (cf.  Chwolson,  Lehrbuchder 
Physik.)  vol.  iii.  p.  654)  for  the  case  of  water. 


t°C. 

Vapour  pressure 
in  mms.  Hg. 

£ 

dt 

A.  observed. 

*i 

*ex 

cals. 

cals. 

cals. 

—  10 

2-093 

0*1611 

6i3'45 

583-I5 

30-30 

0 

50 

4'600 
91-98 

0-329 

4-580 

606-50 
571-66 

575'43 
536-12 

3I-07 

35*54 

100 

760-00 

27-19 

536-50 

496-30          40-20 

150 

3,58l-2 

96-17 

500-79 

456-70          44-09 

2OO 

17,689-0 

243'44 

464-30 

417-70 

47-I3 

yo         A    SYSTEM   OF  PHYSICAL   CHEMISTRY 

If  we  wish  to  use  A,-  instead  of  A  in  the  Clapeyron  equation, 
it  may  be  transformed  into— 


which  yields,  when  VQ  as  compared  to  v±  is  neglected  and  the 
vapour  is  treated  as  a  perfect  gas  — 


A,  =  Ri  -  RT  = 


IV.   Thermodynamic  Expressions  dealing  with  Specific  Heat. 

Let  us  return  to  the  expression  for  the  latent  heat  of 
expansion  /—for  any  system,  homogeneous  or  heterogeneous  — 
namely  — 


/=T(JQo  (equation  (3),  p.  67) 

Differentiating  this  expression  with  respect  to  T,  keeping  the 
volume  constant,  we  obtain  — 


Now,  we  have  already  seen  (p.  66)  that  for  any  system  — 


Combining  these  two  expressions,  we  obtain  — 


This  important  relation  will  be  taken  up  later  in  connection 
with  the  continuity  of  state. 

A  second  important  relation,  dealing  with  the  difference  of 
the  specific  heats  at  constant  pressure  and  volume  respectively 
must  be  considered.  To  get  at  this,  we  must  go  back  to  some 


SPECIFIC  HEAT  RELATIONSHIPS  71 

of  our  earlier  considerations.  If  we  takeany  system  whatso- 
ever from  a  certain  /,  V,  T,  to  another  state  in  which/,  V,  T, 
have  different  values,  we  saw  that  we  had  to  add  a  quantity  of 
heat  which,  for  very  small  transformations,  we  denoted  by  dQ. 
The  absorption  of  this  heat  by  the  system  was  artificially 
divided  up  into  two  consecutive  processes,  viz.  : 

First.  —  The  temperature  rose  by  an  amount  dt>  the  volume 
being  kept  constant.  The  heat  absorbed  amounted  to  Cvdt 
(of  course,  the  pressure  increased  in  this  process,  as  we  see 
when  we  use  the  term  Cv). 

Second.  —  The  system  was  caused  to  expand,  the  temperature 
being  kept  constant  ;  the  requisite  heat  for  this  stage  being  Idv, 
where  /  is  the  latent  heat  of  expansion.  (In  this  process  also, 
the  pressure  underwent  changes  concomitant  with  the  volume 
change.)  These  two  processes  added  together  give  us  the  heat 
absorption  for  the  total  transformation,  the  expression  obtained 
being  — 


which  on  substituting  equation  (3),  p.  67,  for  /becomes  — 


Now  let  us  think  of  the  total  transformation,  when  we  are 
dealing  with  the  same  heat  absorption  </Q,  but  let  us 
artificially  break  up  the  process  into  two  stages  which  differ 
from  the  preceding,  namely— 

First.  —  Let  the  temperature  of  the  system  be  raised  by  an 
amount  dt,  the  pressure  being  kept  constant.  The  heat 
absorption  is  Cpdt,  where  Cp  is  the  specific  heat  of  the 
system  at  constant  pressure.  (In  this  process  naturally,  the 
volume  must  have  undergone  a  concomitant  change.) 

Second.  —  Having  now  the  system  at  the  final  temperature 
value,  suppose  this  to  be  kept  constant,  and  let  us  consider 
that  the  pressure  of  the  system  is  now  allowed  to  rise  by  the 
amount  dp.  This  involves  an  absorption  of  heat  fdpt  when  /' 
is  the  latent  heat  of  pressure  change.  (Naturally  in  this  last 


72         A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

process  the  volume  must  have  changed  concomitantly  with  the 
pressure.)  These  two  processes  added  together  have  been 
assumed  to  give  the  same  total  heat  ;/Q.  For  this  case  there- 
fore we  have  — 


Dividing  across  by  T,  we  obtain  — 

</Q        ,  ,       Cpdt  .  I'  dp 
—  -=  or  do  •  —  —  -  --  --  — 

,j,   ui   uy>  -          T  T 

and  since  d(j>  is  a  complete  differential  we  can  as  usual  write  — 

I/ 

3T\T 


T\T 

acp    si' 

= 


Again,  by  the  First  Law  of  Thermodynamics  for  any  sub- 
stance — 

^Q  =  </U  +  //A 

or  ^Q  =  JU  +  pdv 

:.  d\]  =  dQ  —pdv 
.-.  dU  =  Cpdt  +  I'  dp  —pdv 

and  by  a  simple  mathematical  transformation  — 
d(U  +pv)  = 


Now  since  d\J  is  a  complete  differential,  and  since  on 
completing  a  cycle  the  /  and  v  return  to  their  original  values, 
it  follows  that  d(pv)  is  a  complete  differential,  and  there- 
fore d(\J  -j-  pv)  is  likewise  one,  and  hence  from  the  above 
equation  we  obtain  by  partial  differentiation— 

-Jf+S <•> 


SPECIFIC  HEAT  RELATIONSHIPS  73 

By  combining  (i)  and  (2)  one  finds  — 


Hence  the  equation  — 

</Q 

may  be  put  in  the  form  — 

rfQ  =  C^-Tg)^    ....     (3) 

Now  in  the  alternative  method  of  splitting  up  dQ  into 
parts  containing  dt  and  dv  respectively,  we  have  already  seen 
that  — 


Equations  (3)  and  (4)  must  be  identical,  for  we  considered 
the  same  system  starting  from  the  same  point  and  ending  at 
another  given  point  the  same  for  both,  and  further  we  have 
assumed  that  the  changes  are  reversible  in  both  cases.  Hence, 
equating  (3)  and  (4),  we  get— 


X-«  x^t  f-w-il       v  *^        1  **v  ,         ^r-,1      '-'A'        I  U/U 

or  C^  —  Cv  = 


Now  as  a  purely  mathematical  operation,  if  /  is  a  function 
of  T  and  V,  i.e.  if  p  =/(T,  V),  we  have  a  connection  between 

the  ordinary  differential  -^  (which  means  variation  of  p  with 
T,  while  v  at  the  same  time  varies),  and  the  partial  differential 
^  (which  means  variation  of  p  with  T  when  v  is  kept 

constant). 

This  connection  is  expressed  thus— 

dp      (dp\    .   (dp\      dv 


74         A    SYSTEM   OF  PHYSICAL    CHEMISTRY 

Substituting  this  value  of  J-J  in  the  preceding  equation,  we 
get- 

c       r 
C  ~ 


rr      r      T 

or  Cp-  C.=  * 


Now  a  further  general  mathematical  relation  has  to  be  here 
made  use  of,  namely,  if  we  have  three  variables  x,  y,  and  z 
which  are  mutually  dependent,  then  in  all  cases  we  can 
write  — 

dx 


\    (dy\    (dj\  =  _ 
Jz'\dz)v\dx)y 

Using  the  three  variables  /,  v,  T,  we  can  write  — 


Substituting  this  in  the  preceding  equation,  we  see  that 


and  ,  hereto,        C,-C._T(?  )_.(«)_.    ...    (5) 

And  this  holds  for  any  substance  whatsoever.  It  will  be 
referred  to  later  in  the  next  chapter,  dealing  with  Continuity 
of  State. 


CHAPTER   III 

Continuity  of  the  liquid  and  gaseous  states  from  the  thermodynamic 
standpoint. 

CONTINUITY  OF  STATE  FROM  THE  THERMODYNAMIC 
STANDPOINT. 

IN  the  preceding  chapter  we  have  deduced  several  important 
relations  on  the  basis  of  thermodynamics,  and  we  have  now  to 
see  what  conclusions  they  lead  to  when  applied  to  the  equi- 
librium conditions  which  obtain  in  the  distribution  of  matter 
in  space,  under  varying  conditions  of  temperature  and  pressure. 
It  is  proposed  therefore  to  review  again  the  more  important 
equations  of  state  already  treated  from  the  molecular  kinetic 
standpoint  (Vol.  I.  chap.  ii.).  It  will  be  seen  that  the  intro- 
duction of  these  general  thermodynamic  theorems  considerably 
enlarges  the  number  of  conclusions  which  we  are  able  to 
draw  from  the  respective  equations  of  state,  with  regard  to  the 
behaviour  of  gas  or  liquid  systems.  By  comparing  the  theo- 
retical conclusions  with  actual  experimental  results  we  gain  a 
further  insight  into  the  limits  of  applicability  of  our  equations 
of  state  and  the  assumptions  upon  which  they  rest.  In  this 
way  we  are  able  to  discriminate  to  a  large  extent  between 
conflicting  theories  and  assumptions. 

The  relations  which  we  may  first  consider  are  — 


In  the  first  of  the  above  relations  /  stands  for  the  latent 
heat  of  expansion,  that  is,  the  heat  which  has  to  be  added  to 
a  system  to  keep  the  temperature  constant  while  the  volume 


76         A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

increases  by  unity.  As  already  pointed  out,  this  relation  may 
be  applied  to  homogeneous  and  heterogeneous  systems  alike. 
For  the  present  we  shall  Consider  homogeneous  systems  only, 
i.e.  systems  consisting  entirely  of  gas  or  entirely  of  liquid. 

First  take  the  case  of  a  perfect  gas.     The  characteristic 
equation  for  such  a  body  is  —  • 


We  have  already  seen  (p.  58)  that  for  the  expansion  dv  of  a 
perfect  gas  against  a  pressure  p  the  following  holds  good, 
viz.  :  — 

Idv  = 


No  heat  is  taken  in  to  do  internal  work,  and  therefore  there 
must  be  no  internal  work  to  do.  This  is  a  thermodynamic 
proof  of  the  conclusion  to  which  we  have  already  come  on 
kinetic  considerations,  viz.  that  there  are  no  cohesive  forces 
existing  between  the  molecules  of  a  perfect  gas,  and  therefore 
there  can  be  no  internal  work  done  on  expanding  (cf.  Joule's 
experiment  (p.  27)  ).  Further,  since  for  a  perfect  gas  — 


it  follows  that  on  again  differentiating  with  respect  to  T,  keep- 
ing v  constant, 


=  0 


for  both  R  and  v  are  constant.     Therefore,  from  the  second 
thermodynamic  relation  (p.  75)  considered  it  follows  that—  - 


In  other  words,  the  specific  heat  of  a  perfect  gas  at  constant 
volume  is  independent  of  the  absolute  magnitude  of  the 
volume.  That  is,  suppose  we  consider  one  gram  of  a  perfect 
gas  occupying  a  volume  v,  and  we  raise  the  temperature  i°, 
keeping  the  volume  at  v,  we  require  to  add  a  certain  quantity 
of  heat  Cv.  If  we  consider  the  same  mass  of  gas  at  the  same 
temperature  as  before,  and  at  quite  a  different  volume  z>j  (the 


VAN  DER    WAALS*    EQUATION  77 

pressure  being,  of  course,  different  now),  and  we  raise  the 
temperature  i°,  keeping  v±  constant,  it  will  be  found  that  the 
amount  of  heat  required  is  again  Cy. 

COMPARISON  OF  SOME  OF  THE  EQUATIONS  OF  STATE. 

There  is  no  such  thing  in  nature,  however,  as  a  perfect  gas. 
Let  us  therefore  consider  actual  gases,  and  let  us  suppose  that 
we  can  apply  VAN  DER  WAALS'  EQUATION — 


From  this  we  obtain  by  differentiation 


v  —  b 

therefore  /=      '—-=p-]r~ 

v  —  b  vz 

or  Idv  =pdv  -\  —  ^  dv 

In  words,  for  an  expansion  dv  the  heat  which  has  to  be  added 
to  keep  the  temperature  constant,  viz.  (Idv),  is  equal  to  the 


external  work  done  (pdv)  plus  the  internal  work  done 

against  the  cohesive  force  (  ~g  )•     Thus,  for  a  van  der  Waals' 

gas  the  heat  entry  at  constant  temperature  is  greater  than  the 
heat  which  has  to  be  added  in  the  case  of  a  perfect  gas,  by 
the  amount  representing  the  internal  work  done. 
Again,  on  differentiating  the  equation  — 

dp 


with  respect  to  T  at  constant  vt  we  obtain  — 


.         r  0 

therefore  -=—  =  o 

dv 


A    SYSTEM  OF  PHYSICAL   CHEMISTRY 


i.e.  van  der  Waals'  equation  leads  to  the  conclusion  that  the 
specific  heat  at  constant  volume  of  an  imperfect  gas  (one 
obeying  van  der  Waals'  equation)  is  independent  of  the  volume. 
(The  same  conclusion  was  reached  in  the  case  of  a  perfect 
gas.)  This  conclusion  is,  however,  not  borne  out  in  practice, 
as  the  value  of  C«  is  found  not  to  be  independent  of  volume 
(v).  In  general  as  v  increases  Cv  diminishes,  though  some- 
times the  reverse  is  the  case.  The  following  examples  are 
quoted  by  Kuenen  (Die  Zustandsgleichung^  p.  118,  from 
Reinganum,  Dissertation^  Gottingen,  1899).  In  order  to  deal 
with  different  values  of  v,  the  smaller  limit  for  this  quantity 
refers  to  the  substance  entirely  in  the  liquid  state,  the  higher 
limit  referring  to  the  substance  (at  the  same  temperature)  com- 
pletely in  the  state  of  vapour. 


Substance. 

C,  liquid. 
v  very  small. 

Ctf  vapour. 
v  large. 

Ether  

0-358 

0-346 

Carbon  bisulphide 

o'i6o 

jo'  131  (Regnault) 
(0*105  (Wiedemann) 

Chloroform     . 

0*156 

jo*  140  (Regnault) 
\0'H5  (Wiedemann) 

In  the  case  of  carbon-dioxide,  Dieterici  (Annalen  der 
Physik,  [4]  12,  173,  1903)  states  that  the  Cv  for  small  values 
(i'i  —  1*125  c-c-)  is  °*24  and  that  it  increases  with  increasing 
volume,  reaching  a  maximum  Cv  =  0^34  in  the  neighbourhood 
of  the  critical  volume,  and  then  decreases  with  further  increase 
in  volume.  An  analogous  behaviour  was  observed  in  the  case 
of  isopentane.  Cf.  M.  Reinganum  (Annalen  der  Physik,  18, 
1008,  1905). 

The  RAMSAY  AND  YOUNG  EQUATION  OF  STATE  may  be 
put  in  the  form  — 


and,  as  already  pointed  out,  the  van  der  Waals'  equation  is  a 
special  case  of  this,  so  that  this  relation  leads  to  the  same 
conclusions  regarding  Cv  as  those  already  obtained.  Thus  — 


CLAUSIUS^   EQUATION  79 


/  =  Tl  ^_, )  =  T/i  = 

\OL/v 

or  Idv  —pdv  -f-  F 

a 

;2 


If  we  substitute  ---  for  F(Z')  we  get  the  van  der  Waals'  result. 


Similarly  — 


Now  lef  us  see  to  what  conclusions  the  CLAUSIUS'  EQUATION 
leads  when  treated  in  the  same  manner.  This  equation  it  will 
be  remembered  differs  from  van  der  Waals'  or  Ramsay  and 
Young's  in  that  the  cohesive  force  was  considered  to  vary  with 
the  temperature,  and  at  the  same  time  was  a  more  complex 
function  of  the  density.  The  equation  is  — 

RT  a 


v  —  b      T(v  +  c]v 
In  this  case — 

R  n 

YYJV~  v  —  b  T  T2(z;  +"c)v 

-  _?T  4.         g 

V0     z'  —  ^     ^(^  4"  c)v 

a  a 


-*+«£* 


or  /dfe  =pdv 


In  words,  this  means  that  the  heat  which  has  to  be  added 
to  keep  the  temperature  of  the  gas  (or  liquid)  constant  while  a 
volume  increase  takes  place  is  equal  to  the  external  work 
done  in  expansion  plus  twice  the  internal  work  done  in 
expansion.  In  this  case  therefore  when  the  volume  changes 


8o         A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

the  heat  entry  is  not  equal  to  the  external  +  the  internal  work 
as  in  the  case  of  a  van  der  Waals'  substance,  but  there  is  an 
extra  indraught  as  welk  In  this  case  the  extra  indraught 
happens  to  be  just  equal  to  the  internal  work  done,  viz.— 


Further,  it  is  seen  that — 


tr)C*  \ 
—  -  )  =  a  negative  quantity, 
OV  /T 

i.e.  the  specific  heat  measured  when  the  volume  is  kept  constant 
during  the  i°  rise  should  decrease  when  the  mass  employed  is 
caused  to  occupy  a  greater  volume,  i.e.  by  working  at  different 
pressures.  There  is  no  evidence  of  a  change  of  sign  such  as 
that  actually  exhibited  by  carbon-dioxide  and  isopentane. 
Again,  let  us  take  the  DIETERICI  EQUATION— 


-. 

v  —  b 

which  may  be  written  in  the  form  — 


whence  = 


hence  Idv  —pdv  +  --  .  dv 


i.e.  heat  which  is  required  to  be  added  to  keep  the  temperature 
constant  =  the  external  work  done  plus  a  positive  quantity  due 
to  internal  work,  to  which  no  simple  physical  meaning  can  be 
attached. 


SPECIFIC  HEAT  RELATIONSHIPS  81 

APPLICATION  OF  THE  THERMODYNAMIC  RELATION. 


For  a  perfect  gas  — 

/27  =  RT 

R 


whence 

therefore 


therefore  Cp  —  Cv  =  R,  a  result   to   which   we  have  already 
come  (p.  59,  footnote). 

For  a  substance  obeying  van  der  Waals*  expression  we  would 
have — 

RT     _a 

therefore  (&}  =   JL      therefore  T(|£)  =-^L 

XoT/t,        V  —  O  \OL/V       V  —  b 

Also  to  get  (  ~—  )  we  may  differentiate  van  der  Waals'  equation 
\di/p 

as  given  above.     Remembering  that  p  is  constant,  and  there- 
'dp 


Bp\ 

J  —  o, 


fore  —  o  we  find— 


R  RT     /dzA        2a(dv\ 

~  ~~ 


R 


_        v  —  b 
\dT/p~       RT 


RT        R 


v  —  b'  v  —  b 


RTz'3 

T.C.P. — II. 


82 


A    SYSTEM  OF  PHYSICAL    CHEMISTRY 


Since  the  denominator  is  less  than  unity,  the  expression 
for  Cp  —  Cv  is  greater  than  R,  the  extra  amount  being  due  to 
the  cohesion  effects  between  the  molecules.  The  following 
short  table  gives  some  experimental  values  obtained  for  Cp 
and  Cv  (in  the  region  of  room  temperature).  No  attempt  is 
made  to  compare  these  values  with  the  above  expression, 
owing  to  the  doubt  which  exists  as  to  what  values  should  be 
taken  for  a  and  b  (since  they  are  not  quite  constant)  and 
(Cp  —  Cv)  being  a  small  quantity,  the  calculation  would  be 
sensitive  to  numerical  errors.  The  values  show,  as  might 
have  been  expected,  that  at  this  temperature  oxygen  (O2)  and 
hydrogen  (H2)  approximate  very  nearly  to  perfect  gases, 
whilst  (Cp  —  Cv)  is  in  the  cases  of  ammonia  and  carbon 
dioxide  distinctly  greater  than  R. 


Gas. 

C,,  per  gram. 

Cp  per  gram. 

CCp-C.) 

per  mole. 

R  per  mole 
in  calories. 

Perfect  gas 

_ 

_ 

1-985 

Hydrogen  . 

2'42II 

3-4090 

I'976 

Oxygen 

0-I556 

0-2175 

I-98l 

I-985 

Ammonia  . 

Q'3951 

0-5205 

2-108      • 

Carbon  dioxide 

o*  1669 

0'2l69 

2-20 

The  value  (Cp  —  Cv)  calculated  on  the  Dieterici  equation 
is  very  complicated  and  need  not  be  given  here. 


THE  POROUS   PLUG  EXPERIMENT  OF  JOULE  AND  THOMSON 
AND  THE  PHENOMENA  OF  INVERSION  POINTS. 

This  experimental  investigation  was  carried  out  as  an 
extension  of  the  Joule  experiment  already  alluded  to,  with  the 
object  of  finding  by  much  more  refined  methods  the  diver- 
gencies of  real  gases  from  the  requirements  of  the  perfect  gas. 
The  conditions  of  the  present  experiment,  however,  differ  in 
principle  from  that  of  the  Joule  experiment,  and  it  must  be 
clearly  borne  in  mind  that  the  conclusions  to  be  drawn  from 
each  are  different.  In  the  porous  plug  experiment,  as  the  name 
suggests,  a  gas  was  forced  through  a  resistance  consisting  of  a 


THE  POROUS  PLUG  EXPERIMENT  83 

diaphragm  fitted  with  a  fine  opening,  situated  in  the  axis  of 
a  wider  tube.  A  diagrammatic  sketch  is  given  in  Fig.  49. 
Owing  to  the  resistance  of  the  plug,  the  pressure  was  higher  on 
one  side  than  on  the  other,  and  it  was  shown  by  means  of  a 
sensitive  thermometer  placed  at  the  plug  that  on  the  low- 
pressure  side  the  temperature  was  less  than  that  on  the  high- 
pressure  side.  This  behaviour  was  exhibited  by  all  the  gases 
examined  by  them  with  the  exception  of  hydrogen,  which 
was  found  to  be  at  a  higher  temperature  on  the  low-pressure 
side.  On  passing  through  the  plug,  therefore,  most  gases  are 
cooled,  hydrogen  being  warmed  ;  the  experiment  being  carried 
out  in  the  region  of  atmospheric  temperature.  When,  however, 
the  experiment  is  carried  out  at  a  sufficiently  low  temperature, 
there  is  a  cooling  even  in  the  case  of  hydrogen  on  passing 

p,v,  V, 


FIG.  49. 

through  the  plug.  This  behaviour  suggests  the  existence  of  an 
inversion  point  (or  more  than  one,  as  we  shall  see  later)  for  all 
substances,  at  which  point  there  would  be  neither  cooling  nor 
heating  on  passing  through  the  plug.  Joule  and  Thomson 
found  that  the  cooling,  f.e.  the  fall  in  temperature  produced, 
was  proportional  to  the  difference  in  pressure  on  the  two  sides 
of  the  plug.  That  is  if  —  A/  represents  the  cooling,  then 
-  &t  =  k(p^  — /0))  where  (p±  — A)  'ls  tne  difference  of  pressure 
and  k  is  a  constant  characteristic  of  the  fluid  under  examination 
and  is  the  fall  of  temperature  for  i  atmosphere  difference  in 
pressure.  At  atmospheric  temperature  in  the  case  of  air,  it  was 
found  that  k  =  0*262°;  for  carbon  dioxide  /£=  1-225°.  For 
hydrogen  k  is  negative.  Natanson  found  later  in  the  case 
of  CO2  at  20°  C.  that  k  increases  somewhat  with  the  pressure. 
Joule  and  Thomson  further  found  that  k  is  inversely  propor- 
tional to  T2,  that  is— 


84         A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

Rose-Innes  (Phil  Mag.,  45,  227,  1898)  found  also  that 
the  results  obtained  by  Joule  and  Thomson  for  the  cooling 
(k)  per  i  atmosphere  pressure  difference  could  also  be  re- 
presented by  the  formula — 

..*-; 

It  "may  be  pointed  out  that  the  Rose-Innes  formula,  if  a 
valid  one,  indicates  that  a  point  of  inversion  must  exist,  i.e. 

T>  T> 

when  A  =  —  or  T  =  —  so  that  it  agrees  with  the  observation 
1  A 

already  referred  to  and  obtained  by  Olszweski  in  1901  in  the 
case  of  hydrogen.  Now  the  cause  of  the  thermal  effect  is  the 
divergence  of  the  gases  from  the  perfect  state.  These  effects 
may  be  divided  into  deviations  from — 

(1)  Boyle's  Law.     (Boyle's  Law  states  pv  =  RT.) 

(2)  Joule's  Law.     (Joule's  Law  states  that  U  =  KT  and  is 

independent  of  volume.) 

The  deviations  from  Boyle's  Law  may  be  separately  ex- 
amined, as  has  already  been  done  in  dealing  with  the  pv  X  p 
diagram  of  Amagat  (Vol.  I.  chap.  ii.).  If  we  could  eliminate 
this  we  could  get  the  deviation  from  Joule's  Law.  Consider 
in  the  diagram  of  the  tube  (Fig.  49)  two  points,  A  and  B,  suffi- 
ciently far  removed  from  the  plug  itself  that  the  gas  flow  is 
steady  at  these  points.  Suppose  A  is  on  the  high-pressure  side 
and  suppose  further  that  Uj  is  the  internal  energy  of  i  gram 
of  gas  at  the  point  A,  and  U2  is  the  internal  energy  of  i  gram 
of  gas  at  the  point  B.  Consider  the  block  of  gas  AB.  Sup- 
pose it  to  move  so  as  to  occupy  the  volume  AjBi.  There  is 
the  same  mass  of  gas  in  the  slab  AAX  as  there  is  in  BBlf 
though  of  course  the  volumes  are  different,  since  the  pressures 
are  different.  Suppose  for  simplicity  that  each  slab  contains 
i  gram  of  gas.  Further,  let  AAl  =  Vl  and  BBj  =  V2.  In 
forcing  the  block  of  gas  AB  through  the  tube  containing  the 
plug  the  work  done  on  the  gas  at  the  high-pressure  side  is 
/!#!,  when  /!  is  the  high  pressure.  The  work  done  by  the 
gas  on  the  low-pressure  side  is  p^v^  The  net  work  done  on 
the  gas  equals  /^  — 


THE  IMPERFECT  GAS  85 

This  is  a  gain  in  energy,  since  heat  has  been  neither  added 
nor  subtracted.     The  net  gain  in  energy  is  U2  —  Uj  ; 

therefore  /^  —  /2^2  =  U2  —  Ui 

or  ^        U 


That  is,  in  general  on  both  sides  of  the  plug  we  have  the  same 
value  for  the  expression  (pv  +  U),  or  — 

pv  -j-  U  =  constant. 

If  Boyle's  Law  (pv  =  RT)  and  Joule's  Law  (U  =  KT) 
both  held  gold,  then/z>  -f-  U  would  depend  on  temperature  only  ',  i.e. 
would  be  uninfluenced  by  the  volume  change  which  necessarily 
takes  place  on  passing  from  a  high  to  a  low  pressure.  But 
(pv  -f-  U)  is  shown  to  be  constant  whether  the  gas  laws  are 
obeyed  or  not)  and  hence,  if  they  were  obeyed,  T  would  necessarily 
be  a  constant  likewise,  that  is,  there  would  be  no  Joule-Thomson 
effect  (cooling  or  heating)  on  passing  through  the  plug.  The 
existence  of  the  temperature  change  is  therefore  due  to  one  or 
both  of  the  laws  breaking  down. 

(i)  Deviation  from  Boyle's  Law. 

This  need  not  be  gone  into  here,  as  it  has  already  been  dis- 
cussed (Vol.  I.  chap.  ii.).  It  may  be  recalled  that  with  increase 
of  pressure  up  to  a  certain  value  gases  —  with  the  exception  of 
hydrogen  —  show  themselves  to  become  more  compressible 
than  the  law  required.  In  the  plug  experiment,  therefore,  we 
see  what  the  effect  of  this  will  be,  that  the  product  (pv)  for  a 
given  mass  is  greater  on  the  /<?w-pressure  side  than  the  corre- 
sponding product  for  the  same  mass  on  the  high-pressure  side. 
Here,  in  order  that  pv  may  become  the  same  on  both  sides 
(since  pv-\-\5  is  the  same  on  both  sides)  it  is  necessary  to 
lower  the  temperature  on  the  low-pressure  side,  i.e.  the  gas 
cools  on  passing  through  the  plug.  Thus  from  the  observed 
deviations  from  Boyle's  Law  we  would  expect  a  cooling  on  the 
lower  pressure  side  of  the  plug  except  in  the  case  of  hydrogen.1 

1  Note  that  with  regard  to  the  Amagat  (  (pv)p)  diagram  one  must  be 
careful  not  to  confuse  the  inversion  point  of  the  plug  thermal  effect  above 


86          A   SYSTEM  OF  PHYSICAL    CHEMISTRY 

Further,  we  may  consider  the  case  in  which  the  pressure  is  so 
great  that  for  all  gases  (at  room  temperature)  the  compressi- 
bility is  less  than  that  of  «.  perfect  gas,  i.e.  we  suppose  that  we 
are  at  the  region  of  ascending  curves  on  the  ((pv)p)  diagram. 
In  such  a  case  the  value  of  pv  of  a  given  mass  on  the  high- 
pressure  side  may  be  greater  than  that  on  the  low  pressure 
side,  and  hence  the  temperature  on  the  low-pressure  side 
would  have  to  be  raised  to  bring  the  pv  up  to  the  value  of  the 
other  side.  It  is  conceivable,  however,  that  the  pressures 
might  be  so  chosen  that  the  pv  values  (as  given  on  the  Amagat 
diagram)  are  the  same  on  both  sides  of  the  plug.  This,  how- 
ever, would  not  mean  that  the  temperature  would  remain  the 
same  on  both  sides,  for  the  expression  which  must  be  constant 
is  (U  -\-  pv)i  not/z>  alone. 


(2)  Deviation  from  Joule's  Law. 

According  to  Joule's  Law  the  internal  energy  of  a  given 
mass  should  depend  on  the  temperature  and  not  on  the  volume 
occupied.  Since  in  actual  gases  cohesive  forces  are  present, 
it  follows  that  in  an  expansion,  work  must  be  done  in  drawing 
the  molecules  apart,  and  therefore  at  the  larger  volume  the 
given  mass  contains  a  greater  value  of  U  than  at  the  smaller 
volume,  and  hence  if  U  is  to  be  constant  on  both  sides  of  the 
plug  we  must  decrease  the  temperature  on  the  larger  volume 
side,  i.e.  in  the  low-pressure  side.  This  should  hold  for  all 
gases. 

We  might  summarise  the  thermal  effects  to  be  expected 
on  the  low-pressure  side  of  the  plug  owing  to  deviations  from 
the  Gas  Laws  as  follows  :  — 

Deviations  from  Boyle's  Law.  —  Heating  or  cooling  pro- 

referred  to,  with  the  series  of  temperatures  for  which  for  a  given/  the  ex- 
pression pv  is  a  minimum.  (At  such  points  it  is  true  that  Boyle's  Law  is 
momentarily  obeyed,  and  if  the  plug  experiment  required  the  expression  pv 
to  be  the  same  on  both  sides,  then  we  could  calculate  from  Amagat's 
(  (pv}p}  diagram  what  difference  of  pressure  is  required  at  a  given  tem- 
perature to  cause  no  thermal  effect.  The  expression  which  holds  for  both 
sides  of  the  plug  is,  however,  pv  +  U  =  constant).  There  is  no  connec- 
tion between  the  porous  plug  inversion  point,  and  the  (  (pv)p)  diagram 
"  inversion  "  points. 


THE  IMPERFECT  GAS  87 

duced  according  to  the  temperature  and  actual  absolute 
pressure  worked  at. 

Deviations  from  Joule's  Law.  —  Cooling  in  all  cases.  The 
observed  phenomena  will  be  the  resultant  of  these  simultaneous 
effects. 

The  Joule-Thomson  porous  plug  experiment  has  received 
an  important  technical  application  in  that  it  is  the  basis  of  one 
of  the  methods  used  in  the  liquefaction  of  gases.  (See  Travers' 
Experimental  Study  of  Gases.) 

We  may  now  treat  the  porous  plug  thermal  phenomena  from 
a  somewhat  more  quantitative  thermodynamic  standpoint.  In 
the  first  place  it  must  be  pointed  out  that  we  are  dealing  with 
an  irreversible  phenomenon,  since  we  cannot  make  the  gas 
retrace  its  path  from  low  to  high,  the  pressure  difference  being 
a  finite  one,  and  the  heat  is  therefore  dissipated.  We  know 
already  that  while  the  First  Law  may  be  applied  to  such  pro- 
cesses, the  Second  Law,  and  any  deductions  naturally  which 
involve  the  Second  Law,  cannot  be  applied.  This  Second 
Law  holds  good  for  reversible  processes  alone.  We  must 
therefore  not  go  beyond  the  limits  of  the  First  Law  and  the 
conclusions  which  are  based  upon  it. 

We  have  already  seen  that  the  expression— 


may  be  written  in  several  forms,  one  of  which  is 


when  Cp  is  the  specific  heat  of  the  fluid  at  constant  pressure, 
Also  for  a  substance  which  undergoes  a  volume  change — 


p 
Also  adding  d(pv)  to  each  side — 


<U  +/tf)  =  Cpdt  —  T(  ^  )  dp  —pdv  -\-pdv  +  vdp 
\/p 


88         A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

since  (U  +/z>)  has  the  same   value   on   both  sides  of  the 
plug. 

This  expression  should  hold  good  in  all  cases.  Now,  if 
we  happen  to  be  at  the  temperature  at  which  there  is  no 
change  of  temperature,  i.e.  A^  =  o,  and  since  dp  is  not  zero, 
we  have  — 


as  the  criterion  of  the  inversion  point,  this  denoting  the  tem- 
perature at  which  the  thermal  effect  is  nil.  We  can  see, 
perhaps,  a  little  more  clearly  the  physical  meaning  of  this 
expression  if  we  divide  across  by  v.  The  expression  is  then— 

T/dzA  i(dv\       i 

"        •' 


"       ' 


The  coefficient  of  thermal  expansion  at  constant  pressure  is 

-(  —  )  ,  and  hence  the  inversion  point  is  denned  as  the  tem- 
zAal  /  p 

perature  at  which  the  coefficient  of  expansion  of  the  fluid  at 
constant  pressure  is  equal  to  the  reciprocal  of  the  absolute 
temperature. 

THE  CRITERIA  OF  A  PERFECT  GAS. 

If  there  is  no  temperature  change  in  the  Joule-Thomson 
experiment,  the  following  relation  must  hold  :— 


p 
This  will  be  satisfied  by  writing — 


where  fi(p)  is  an  integration  constant  independent  of  T.  A 
perfect  gas  will  satisfy  this  as  a  special  case,  for  a  perfect  gas 
has  no  plug  thermal  effect  at  any  temperature.  Further,  if 
there  is  no  change  in  temperature  in  the  Joule  experiment,  we 
have  (p.  68)— 


THE   PERFECT  GAS  89 

This  will  be  satisfied  by  the  expression  — 


where  f£v)  ls  an  integration  constant  independent  of  T. 
Again,  a  perfect  gas  must  satisfy  this  as  a  special  case.  Other 
substances  might  satisfy  one  or  other  of  the  criteria  v  =  T/i(^) 
and  p  =  Tf2(v)t  but  to  satisfy  both  the  substance  must  be  a 
perfect  gas.  For  suppose  we  have  a  substance  for  which  the 
relations  hold  good  simultaneously,  viz.  :  — 


then  for  this  substance  we  must  have  — 


In  order  that  the  right-hand  side  may  really  be  /,  it  is  evident 

k  k 

that  f2v  must  be  -,/j/must  be  -,  k  being  a  constant  the  same 

in  both.    For  this  substance  therefore  which  satisfies  the  above 
relations,  the  two  preceding  relations  take  the  form  of — 

iJ£L\ 

v    I   i.e.  both  expressions  become 
_  kT  j  identical,  giving — 

=  7  J 
pv  —  kT 

which  is  the  equation  characteristic  of  a  perfect  gas. 


THE  USE  OF  "  POROUS  PLUG  "  INVERSION  POINTS  IN 
TESTING  PROPOSED  EQUATIONS  OF  STATE.* 

The  way  of  setting  about  this  problem  is  to  inquire  if  the 
equations  which  have  been  proposed  for  real  (imperfect)  gases 

1   Cf.  A.  W.  Porter,  Phil.  Mag.,  series  [6],  p.  554,  vol.  11,    1906; 
series  [6],  p.  888,  vol.  19,  1910. 


90         A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

will  indicate  the  existence  of  an  inversion  point,  if  the  gas  be 
passed  through  a  porous  plug.  Will  a  gas  which  obeys  van  der 
Waals'  expression,  for  example,  show  this  behaviour  ? 

(a)  Investigation  of  van  der   Waals1  Equation. 

Instead  of  writing  this  in  the  usual  form,  it  is  more  con- 
venient to  make  use  of  the  reduced  form,  i.e.  pressures,  volumes, 
and  temperatures  will  be  expressed  as  fractions  a,  j3,  y  of  their 
critical  values.  The  results  will  be  the  same  for  every  fluid 
obeying  this  equation  (cf.  the  section  dealing  with  correspond- 
ing states  (Vol.  I.  chap,  ii.) ).  The  reduced  van  der  Waals' 
equation  is — 


Now  we  have  already  seen  that  the  equation  characteristic 
of  an  inversion  point  is — 


or,  writing  this  in  the  reduced  form  also,  we  obtain — 


>-»- 


dy 

By  differentiating  (i)  with  respect  to  j8,  and  substituting  in 
(2),  we  obtain  for  an  inversion  point  the  relation— 

-8y        6(3/3- i) _ 
3J8-  i4         P~ 

From  this  it  follows  that  the  inversion  temperature  (on  the 
reduced  scale)  is  given  by — 


From  this  and  the  previous  equation,  y  can  be  eliminated  with 
the  result  — 

«  =  (4) 


THE   INVERSION  POINT  91 

This  formula  connects  the  reduced  pressure  and  volume 
which  correspond  to  an  inversion  point.  The  simplest  mode 
of  calculation  is  to  obtain  y  by  equation  (3)  for  a  series  of 
assumed  values  of  jS,  and  then  to  calculate  a  by  means  of 
equation  (4)  for  the  same  values  of  j3.  Working  in  this  way 
it  is  found  that  on  the  y,  a  diagram  (Fig.  50)  the  curve  obtained 
is  a  parabola.  This  means,  in  the  first  place,  that  for  a  given 
fluid  there  is  a  continuous  series  of  inversion  points  possible 
(corresponding  to  different  values  of  /3),  up  to  a  certain  tem- 
perature, above  which  there  is  no  inversion;  and,  still  more 


Warmind   Redion 


Warming  Reckon 


x  Nitrogen 

o  Carbon  dioxide 


o 
pressure. 


/x 


FIG.  50. 

striking,  there   are  two  temperatures  corresponding   to  each 
pressure  at  which  inversion  can  occur.1 

In  the  diagram  (Fig.  50)  the  dotted  parabolic'curve  gives 
the  position  of  the  inversion  points  as  deduced  from  van  der 
Waals'  equation.  A  line  drawn  perpendicular  to  the  a  ordinate 
cuts  the  parabola  at  two  points,  these  points  being  the  inver- 
sion temperatures  for  the  given  reduced  pressure. 

1  The  quantitative  relations  are  summarised  by  Porter  as  follows  : — 

(1)  For  all  pressures  from  zero  to  nine  times  the  critical  pressure  there 
are  two  inversion  temperatures  which  may  range  from  a  little  below  the 
critical  temperature  to  about  6"j  times  the  critical  value. 

(2)  At  pressures  higher  than  nine  times  the  critical  value  there  is  no 
inversion  point. 


92          A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

(b)  Investigation  of  Dietericis  Equation. 

This  equation,  applied  by  Dieterici  to  the  case  of  iso- 
pentane,  is  written — 

p(v  —  b)  = 
or  in  the  reduced  form 


The  inversion  points  corresponding  to  this  equation  are  given 
by  the  relation  — 


This  also  gives  a  parabolic  curve  for  inversion  points,  which 
is  shown  as  a  full  line  in  the  diagram  already  referred  to.  The 
curve  obtained  by  Dieterici's  and  van  der  Waals'  equations 
respectively  are  of  the  same  type,  but  at  the  same  time  they 
differ  so  markedly  that  it  should  be  possible  to  discriminate 
between  them  as  regards  their  validity  on  comparing  the  results 
with  those  based  on  experimental  data.  Unfortunately  such 
data  are  very  scanty.  Porter  (loc.  dt.)  has  calculated  some 
inversion  points  in  the  case  of  nitrogen  as  far  as  Amagat's 
data  permitted.  A  similar  calculation  was  carried  out  in  the 
case  of  carbon  dioxide.  In  the  diagram  the  experimental  points 
for  nitrogen  are  denoted  by  crosses,  those  for  carbon-dioxide 
being  denoted  by  circles.  In  neither  case  are  there  sufficient  data 
available  to  permit  the  following  of  the  parabolic  curve  over 
a  wide  range.  In  the  case  of  nitrogen,  however,  the  turning 
point  indicating  the  highest  temperature  at  which  inversion  is 
possible  is  clearly  seen.  It  is  important  to  note  that  while 
the  values  for  nitrogen  lie  on  the  upper  part  of  a  parabolic 
curve,  those  for  carbon  dioxide  lie  on  the  lower  part.  A 
glance  at  the  diagram  is  sufficient  to  show  how  much  better 
Dieterici's  equation  reproduces  the  experimental  values  than 
does  van  der  Waals'  equation.  On  the  basis  of  the  porous 
plug  experiment  the  conclusion  is,  therefore,  that  Dieterici's 
equation  is  more  in  agreement  with  the  observed  behaviour  of 
fluids  than  is  van  der  Waals'. 


MAXWELL'S  RELATION 


93 


HETEROGENEOUS  SYSTEMS  CONSISTING  OF  SATURATED 
VAPOUR  IN  CONTACT  WITH  LIQUID. 

The  Equality  of  the  Segments  of  the  Hypothetical  Isotherm  in 
the  Heterogeneous  Region. 

According  to  the  Second  Law  of  Thermodynamics  the 
external  work  done  by  a  system  in  passing  isothermally  and 
reversibly  from  the  initial  to  the  final  stage  is  independent  of 


FIG.  51. 

the  path  followed.  Let  us  apply  this  principle  to  the  con- 
sideration of  the  passage  from  the  completely  gaseous  (volume 
v-^  to  the  completely  liquid  state  (volume  v0),  this  change 
being  supposed  to  take  place  isothermally.  Referring  to  the  pv 
diagram  (Fig.  51)  already  given,  it  is  evident  if  we  consider 
the  region  of  saturation  that  we  can  pass  from  v±  to  VQ  by 
either  of  two  ways ;  namely,  along  the  horizontal  constant 
pressure  line,  which  is  the  actual  passage  followed  by  the 
system,  or  on  the  other  hand,  by  the  hypothetical  isotherm 
suggested  by  James  Thomson,  and  reproducing  the  van  der 


94         A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

Waals'  equation.  Following  the  first  path,  namely,  the  hori- 
zontal AB,  the  external  work  done  in  compressing  is  repre- 
sented by  the  expression  p(v±  —  VQ),  where  p  is  the  vapour 
pressure.  This  work  is  represented  by  the  area  ABCD. 
Following  the  second  path,  namely  BEGFA,  the  work  done 

/••"o 

is  I    pdv,  where  /  is  no  longer  constant  but  varies  continually 
J  ^ 

throughout  the  volume  change.  This  work  term  is  represented 
in  the  diagram  (Fig.  51)  by  the  area  BEGFADC,  and  it  is 
evident  that  if  BEGFADC  is  equal  to  ABCD,  it  follows  that 
the  two  areas  BEG  and  GFA  are  equal,  that  is : — 


f 

—    VQ)  =  \ 

J 


This  was  first  pointed  out  by  Maxwell.  The  conclusion  is 
probably  correct,  but  it  must  not  be  forgotten  that  the  second 
path  followed  is  a  hypothetical  one  never  yet  realised  in 
practice — at  least  between  E  and  F. 

The  Method  of  applying  van  der  Waals'  Equation  to  the 
Heterogeneous  System,  Vapour- Liquid. 

It  has  been  already  pointed  out  that  van  der  Waals' 
equation  applies,  i.e.  reproduces  experimental  facts  at  least 
approximately,  for  the  homogeneous  system  consisting  either 
of  liquid  or  of  gas.  By  the  introduction  of  the  Maxwell 
assumption  considered  above,  it  is  possible  to  use  van  der 
Waals'  equation,  to  obtain  information  regarding  pressures  and 
volumes  of  heterogeneous  systems  in  the  following  way.  The 
two  points  A  and  B  both  lie  on  the  hypothetical  and  the  real 
isotherms.  To  each  of  these  points  we  can  apply  the  van  der 
Waals'  equation.  Further,  the  points  A  and  B  represent  the 
values  which  are  the  limits  of  integration  in  the  expression 

/  lpdv.     We  can  therefore  apply  van  der  Waals'  equation  to 

carry  out  this  integration  by  writing  p  as  a  function  of  vt 

namely — 

RT     _a 

-b      tf 


PRESSURE  OF  SATURATED    VAPOUR  95 


Hence  (V  =  *T  log 

J  V 


+  *  -  * 

Q  —          '      Z/j  VQ 


a  and  £  being  regarded  as  constants. 

But  according  to  Maxwell's  assumption 


or 


The  point  A  is  characterised  by  the  equation— 


The  point  B  is  characterised  by— 

-*)  =  RT  •  •  •  •  (3) 


From  these  three  equations  it  is  possible  to  find,  for 
example,  vapour  pressures  or  specific  volumes  of  liquid  and  of 
saturated  vapour  as  functions  of  the  temperature. 

Expressions  for  the  Pressure  of  Saturated  Vapour  (in  contact 
with  Liquid)  and  the  Heat  of  Vaporisation. 

In  the  section  dealing  with  the  continuity  of  state  from 
the  kinetic  standpoint,  we  have  considered  several  expressions 
of  this  kind  in  some  detail  (Vol.  I.  chap.  ii.).  The  problem 
still  deserves  a  little  further  discussion.  The  most  important 
relation  between  latent  heat  of  vaporisation  and  pressure  is 
that  deduced  on  the  basis  of  the  First  and  Second  Laws  of 
Thermodynamics,  and  known  as  the  Clapeyron  equation, 
which  may  be  written,  on  the  assumption  that  the  vapour 
obeys  the  gas  laws,  in  the  form  — 


in  which  A,ft  represents   the  molecular   heat  of  vaporisation, 


96         A    SYSTEM   OF  PHYSICAL    CHEMISTRY 

R  having  the  value  1*985  calories.  This  can  be  rewritten  in 
terms  of  the  concentration  C  of  the  saturated  vapour  by 
applying  the  gas  law  in  4he  form  /  —  CRT,  namely— 


a  i 

The  left-hand  expression  denotes  the  internal  molecular  latent 
heat  of  vaporisation.  As  measurements  show  this  heat 
varies  in  a  continuous  and  gradual  manner  with  temperature  ; 
we  can  therefore  write  — 

Am  -  RT  =  AO  +  a0T  +  ft/I*  +  y0T3  +  etc. 
Employing  this  to  integrate  the  above  equation,  one  obtains— 


Where  /  is  an  integration  constant.  We  can  easily  transform 
this  back  into  vapour  pressure  terms  by  putting  /  =  RTC, 
whereby  we  obtain  — 


+  **+logR 

The  last  two  terms  may  be  added  and  considered  as  a  single 
constant  — 

*  +  logR 
L/o  =  - 

2-3023 

This  term  C0  is  called  by  Nernst  (Applications  of  Ther- 
modynamics to  Chemistry]  "  the  chemical  constant  "  of  the 
substance  in  question.  It  must  be  remembered  that  at 
ordinary  temperatures  the  gas  laws  only  hold  approximately 
for  saturated  vapour.  At  lower  temperatures,  however,  the 
application  of  the  gas  laws  becomes  increasingly  more  valid. 
For  this  reason,  therefore,  Nernst  and  his  pupils  have  in  recent 
years  carried  out  a  considerable  number  of  very  accurate 
vapour  pressure  determinations  at  temperatures  considerably 
below  o°  C.  Details  of  these  will  be  found  in  his  book 
already  referred  to  (Applications  of  Thermodynamics  to 


THE  ''CHEMICAL   CONSTANT 


97 


Chemistry],  and   also  J.  T.  Barker  (Zeit.  fur  physik.  C/iew., 
71,  235,  1910). 

The  calculation  of  /  or  C0  can  be  of  course  carried  out 
directly  by  substituting  a  number  of  experimentally  deter- 
mined values  for  log  p  in  the  vapour  pressure  equation  given 
above.  Nernst  has,  however,  modified  this  form  of  procedure 
in  several  ways.  It  would  be,  however,  outside  the  scope  of 
this  book  to  go  further  into  this  point.  Full  details  are  given 
in  Nernst's  Applications  of  Thermodynamics  to  Chemistry. 
The  following  table  contains  a  summary  of  the  values  of  the 

chemical  constant         — - —  (Nernst,  loc.  cit..  p.  75). 
2*3023 


Substance. 

Chemical 
constant  per 
mole. 

Substance. 

Chemical 
constant  per 
mole. 

Hydrogen   . 
Methane 
Nitrogen     . 
Oxygen 
Carbon-monoxide  . 
Chlorine 
Iodine    .... 
Hydrochloric  acid 
Nitric  oxide 

2-21 

ri 

2'8 

3'6 
3-0 
4-0 
3-0 

app.  3  '  7 

Carbon-dioxide    . 
Carbon-bisulphide     . 
Ammonia 
Water        .... 
Carbon-tetrachloride 
Chloroform 
Benzene    .... 
Ethyl  alcohol 
Ether  

3-2 

3;i 

3-7(3-6) 

3*i 

3*2 

3'i 
4'i 

"V  3 

Nitrous  oxide   . 
Sulphuretted  hydrogen 
Sulphur-dioxide     . 

3  3 
3*o 

3*3 

Acetone     .... 
Propyl  acetate 

3*7 

3'8 

1  Nernst's  Text-book,  English  translation  of  the  6th  German  edition, 
gives  C0=  i'6  for  H2. 

As  a  general  rule  the  value  of  the  chemical  constant  is 
about  3. 

Another  interesting  relation  is  that  known  as  Troutoris 
Law.  According  to  this  law  the  molecular  latent  heat  of 
vaporisation  divided  by  the  boiling  point *  is  a  constant.  The 
limits  of  applicability  will  be  seen  from  the  following  table 
(Louguinine's  data,  Winkelmanrts  Handbuch,  vol.  iii.). 

1  In  absolute  units. 


T.P.C. — II* 


98 


A    SYSTEM  OF  PHYSICAL    CHEMISTRY 


Substance. 

- 

Boiling  point. 

M\ 

T 

Methyl  ethyl  acetone  .      . 
Diethyl  acetone 
Dipropyl  acetone  . 
Acetal  

78-68°  C. 

IOI'08 

I43'52 
102-91 

21-25 
20-90 
20-73 
20*78 

Octane 

I2H*  7 

20*28 

Aniline                   • 

I  84  '  24 

21  *22 

O.  Toluidine    .... 
Nitro-benzene  .... 
Acetonitrile      .... 
Pyridine      

210-6 

81-54 
115-51 

21*55 
20-70 
19-74 
20*12 

Ethyl  alcohol   .... 
n.-Propyl  alcohol  .      .      . 
iso-Butyl  alcohol    . 

78-20 
96-1 
107-53 

26-39 

26-59 
26-12 

Acetic  acid       .... 
Propionic  acid 

119*2 

141-05 

I3-74 
16-34 

The  "  normal "  value  for  the  constant  is  about  20-7,  and  a 
large  number  of  different  substances  approximate  fairly  closely 
to  this.  On  the  other  hand,  substances  such  as  the  alcohols 
and  acids,  which  are  known  to  be  polymerised  in  the  liquid 
state,  give  different  values  for  the  constant.  The  law  therefore 
is  not  general.  Although  put  forward  in  the  first  instance  as 
an  empirical  relation,  it  has  a  certain  amount  of  theoretical 
basis  from  the  standpoint  of  van  der  Waals'  theory  of  corre- 
sponding states.  The  reasoning  is  as  follows  :— 

Starting  from  the  Clapeyron  equation — 


and  rewriting  the  T,  /,  and  v  terms  in  reduced  units — 
P  v  T 


this  expression  becomes  — 


z>c   da 


T 


Now  if  we  consider  one  gram-mole  of  each   of  a  series  of 


TROUTOWS  LAW  99 

substances  the  molecular  latent  heat  becomes  MA.  Since  we  are 
dealing  with  the  same  number  of  molecules  in  all  cases,  R  is 

the  same,  and  the  expression   —^  will  be  the  same  for  all 

AC 

™  -n 

(namely,  -7^—  according  to  van  der  Waals'  equation,  or          • 

in  actual  cases).  Further,  if  we  are  comparing  the  substances 
at  corresponding  temperatures,  a,  j8,  and  y  will  be  the  same  for 
all  substances,  and  therefore  at  corresponding  temperatures  the 

expression  -™—  will   be   the  same  for  all   substances.     Now 

Guldberg  (Zeitsch.  physik.  Chem.,  5,  374,  1890)  has  pointed  out 
that  the  ordinary  boiling  points  of  liquids  —  under  atmospheric 
pressure  —  are  practically  corresponding  temperatures  ;  the 
boiling  points  being  approximately  two-thirds  of  the  critical 

temperature,  i.e.  y  =  f  .     Hence  at  the  boiling  point  -^-  should 

be  constant  for  all  substances.     This  is  Trouton's  Law. 

A  further  semi-empiric  expression  for  MA  7  has  been  given 
(Nernst,  Applications  of  Thermodynamics  to  Chemistry,  p.  103, 
or  Theoretical  Chemistry^  English  translation,  6th  German 
edition,  p.  273),  namely— 


in  which/!  and/2  denote  vapour  pressures  corresponding  to 
T!  and  T2,  two  temperatures  which  differ  by  so  small  an 
amount  that  their  geometrical  and  arithmetical  means  may 
for  practical  purposes  be  said  to  equal  one  another.  This 
mean  temperature  is  the  one  to  which  A  corresponds.  This 
formula  gives  in  fact  values  which  agree  with  the  direct 
measurements  ;  in  general  the  heats  of  vaporisation  calcu- 
lated with  its  aid  are  more  accurate  than  those  determined 
calorimetrically.  In  the  following  table  are  given  the  values 
of  the  boiling  point  T0  (on  absolute  scale),  and  molecular 
latent  heat  of  vaporisation  calculated  by  the  above  expression  ; 

1  Nernst  uses  A.  to  denote  molecular  latent  heat.     We  have  used  it  to 
denote  latent  heat  per  gram. 


ioo       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

with  the  exception  of  the  case  of  hydrogen,  where  Dewar's 
experimental  value  is  employed.     The  third  column  gives  the 

value  of  •=-- ,  which  is  Crouton's  expression.     It  will  be  seen 

AO 
that  this  value  is  by  no  means  constant. 


MA 

Substance. 

TO- 

in  calories. 

TO 

9-5logT0-o-oo7T0. 

Hydrogen 

20-40 

248 

I2'2 

I2'3 

Nitrogen  . 

77'5 

1362 

I7'6 

Air     .      . 

86-0 

1460 

I7-0 

I7'8 

Oxygen    . 

90  '6 

1664 

18-0 

Ether 

307 

6466 

21'  I 

21-5 

Carbon  bisulphide 

319 

6490 

20'4 

21-6 

Benzene  .      .      . 

353 

7497 

21-2 

21-7 

The  expression  occurring  in  the  last  column  is  given  by 
Nernst,  who   calls   it   the   revised   rule  of  Trouton,  i.e.    the 

MA  . 
expression  Tp—  is  not  to  be  regarded  as  a  constant,  but  as  a 

AO 
function  of  the  temperature  according  to  the  relation — 

-~-  =  £j  log  T0  —  £2T 
Ao 

where  k±  and  £2  are  9'5  and  0*007  respectively  for  all  sub- 
stances. In  this  case,  however,  the  polymerised  substances 
do  not  show  agreement.  The  further  relations  connected 
with  the  pressures  of  saturated  vapours  will  be  taken  up 
later  from  the  standpoint  of  Nernst's  Third  Law  of 
Thermodynamics. 

Bakkcr's  Equation. 

In  1888  (Dissertation,  Schiedam)  G.  Bakker  put  forward  the 
following  relation— 


where  K  =  internal  pressure  or  cohesive  force  per  unit  area 
across  any  section  in  the  interior  of  the  fluid.     ¥^dv  represents 


BAKKER -S  EQUATION  ici 

therefore   the  internal  work  done  when  the  system  expands 
by  dv. 

A  =  latent  heat  of  vaporisation  per  gram, 
z/j  =  volume  of   i  gram  of  vapour  \  at  the  same  tem- 
VQ  —  volume   of    i    gram    of   liquid  J        perature. 
p  =  vapour  pressure. 

We  may  write  this  expression  in  the  approximate  form — 

A  =  /W~ 
in  which  we  have  neglected  z/0  compared  to  z^  and  have  put 

T>  rr\ 

pv±  =  --JTT-  as  a  first  approximation  (M  =  molecular  weight  of 

vapour). 

Or  calling  A^  the  internal  latent  heat  per  gram — 

Kdv 


Bakker  integrated  this  expression  on  the  assumption  that 
K  may  be  represented  as  a  function  of  z/,  according  to  the 

expression  K  =  — ,  where  A  is  a  constant.  This  leads  to 
the  equation — 

A^  =  A(  — ),  or  approximately  — 

\z>0      V  3  vQ 

It  may  be  noted  that  Bakker  deduced  this  expression  with- 
out reference  to  van  der  Waals'  equation.     In  van  der  Waals' 

equation  K  =  — ,  so  that  the  A  in  Bakker's  equation  would 
become  identical  with  a  provided  K  were  really  represented 
by  ~2  that  is  provided  a  were  independent  of  temperature.  Bakker 

himself  (Zeitsch.  physik.  C/tem.t  12,  670,  1893)  has  shown  the 
connection  between  A  and  a,  namely— 


102       A    SYSTEM   OF  PHYSICAL    CHEMISTRY 

Let  us,  however,  regard  a  as  independent  of  temperature, 
as  a  first  approximation.  Call  it  at  at  the  temperature  of  the 
vaporisation  considered.  To  illustrate  how  far  Bakker's  re- 
lation in  the  form  — 


applies  in  practice,  a  table  due  to  J.  Traube  (Annalen  der 
Physik.,  [4]  8,  300,  1902),  is  given  below.  It  will  be  observed 
that  ac  (the  value  of  van  der  Waals'  constant  at  the  critical 
temperature)  is  greater  than  at  (the  value  of  the  same  "  con- 
stant "  at  the  temperature  of  vaporisation),  and  hence  Traube 
gives  two  series  of  calculated  values  for  the  latent  heat. 
The  at  values  were  obtained  by  Traube  from  the  abbreviated 
van  der  Waals'  equation  — 

a  (v-b}  =  RT 

Traube  made  use  of  the  values  of  bt  (loc.  cit.,  p.  284; 
ibid.)  5,  552,  1901)  obtained  from  the  van  der  Waals'  equation 
at  two  slightly  different  temperatures  in  the  region  required. 
The  values  of  ac  are  those  calculated  by  Guye  (Arch.  Science 
phys.  et  natur.,  Geneve^  $,22,  1900)  — 

27 


The  column  headed  A  calc.  I.  refers  to  ac',  A  calc.  II.  refers 
to  at.  It  may  be  pointed  out  that  the  term  ac  has  a  much 
more  definite  significance  than  at  since  the  former  is  obtained 
from  the  critical  values  of  P  and  T  alone,  whilst  the  latter 
depends  on  the  more  or  less  arbitrary  value  of  bt  which  holds 
good  for  a  certain  temperature  arbitrarily  chosen.  The  values 
of  at  are  therefore  not  as  comparable  with  one  another  as 
the  ac  values. 


BAKKEKS  EQUATION 


103 


ro 

cj 

CJ 

If 

If 

hH      "•> 

£H   ^ 

1  y 

Substance. 

i 

1 

'<? 

'f* 

5  1 

|| 

If 

jpa! 

Mercury    .... 

360-0 

I4-2 

15-72 

_ 

8-68 

_ 

14,660 

14,540' 

[so-pentane    . 

28-0 

117-9 

18-20 

11-23 

4,340 

2,9IO 

6,000 

n.-Hexane 

69*0 

IO4T 

139-8 

24-58 

i5'37 

3,340 

— 

Ethyl  ether    .      .      . 

34'8 

79'3 

106-4 

1  7  '44 

10-56 

4,590 

3,020 

6,260 

Chloroform    . 

60-9 

84-5 

14-71 

— 

4,880 

— 

6,985 

Carbon  tetra-chloride 
Carbon  bisulphide     . 

76-2 
46-2 

78-1 

103-7 
62-1 

19-20 

11-20 

12-04 

5,185 

5,oio 

3,510 

7,13° 
6,600 

Benzene    .... 

80-25 

72-1 

96-2 

18-36 

11-13 

3.510 

7,29° 

VIethyl  formate   . 

32-9 

47-6 

62-7 

II-38 

5,'oio 

6,970 

Ethyl  acetate 

75'9 

78-7 

106-0 

20-47 

11-79 

5,380 

3^390 

7,640 

Siitrogen  .... 

-  194-4 

33'2 

i  '35 

1,140 

1,620 

Sulphur  dioxide  . 

lO'O 

— 

43'9 

6-61 

— 

4,i75 

— 

6,090 

ithyl  alcohol 

78-1 

—  . 

62-3 

15-22 

— 

6,620 

— 

9,440 

Acetic  acid     . 

119-2 

48-0 

63-8 

17-60 

8-29 

5,39o 

3,930 

7,470 

Water       .... 

100 

15-5 

18-9 

577 

3-29 

8,190 

4,980 

9,660 

It  will  be  noted  that  the  agreement  between  observed  and 
calculated  values  is  not  good,  the  observed  values  being  greater 
than  the  calculated,  and  rather  remarkably,  the  discrepancy  is 
greater  in  the  case  of  the  at  values  than  in  those  calculated 
from  the  ac  values.  Traube  discusses  these  discrepancies,  but 
it  would  be  outside  our  present  purpose  to  follow  him  further. 
It  may  be  noted  that  in  the  case  of  mercury  the  value  of  A 
calculated  from  at  agrees  well  with  the  observed  value,  thus 
pointing  to  the  possibility  that  in  the  case  of  this  substance 
the  constant  a  is  in  reality  very  nearly  independent  of 
temperature.  The  same  is  approximately  true  for  bromine, 
given  in  a  later  paper  by  Traube,  loc.  #/.,  namely,  A  observed, 
7296  calories;  calculated,  6620  calories.  The  same  thing 
holds  good  for  zinc  and  cadmium  (and  perhaps  for  other 
metals  as  well),  viz. : 

1  This  is  not  the  figure  given  by  Traube  in  1902,  but  the  later  value 
given  by  him  in  Zeit.f.  anorg.  Ghent.,  34,  423,  1903. 

The  term  a  is  given  in  (liter) 2  X  atmosphere  because  —  has  the  dimen- 
sions of  a  pressure,  and  therefore    -  is  energy,  so  that  a  —  energy  x  volume, 


104        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 


A.  observed 

a 

- 

per  gram-molecule. 

V 

Zinc  .... 

25,500 

25,450 

Cadmium 

23,480 

23,450 

For  sulphur,  however,  the  discrepancy  is  marked  :  A  ob- 
served, 23,170  calories  per  gram-molecule  ;  —  -f-  RT  =  32,300 

calories.  It  may  be  noted  that  in  Traube's  table  of  liquid 
substances  already  given,  water  and  the  alcohols  which  are 
known  to  be  considerably  associated  do  not  appear  to  behave 
in  any  characteristic  manner  different  from  other  liquids  as 
regards  the  values  of  A  calculated  and  A  observed.  This  is 
rather  surprising. 

a,        RT 
Bakker's   equation    A  =  -  -  +  -^r   may    be  put    into    an 

alternative  form  by  substituting  an  expression  containing  b  by 
the  aid  of  van  der  Waals'  equation.  Thus,  according  to 

van  der  Waal's  equation,  since  — g  is  simply  K,  the  cohesive 
pressure, 


T} 

where  R'  refers  to  i  gram  and  may  be  written  — ,  R  being 
1*985  cals. 


hence 
Now 


R'T 

—  —j 

v  —  b 


—pdv 


=  R'T  log  ^  _ 


=  p  ! 

J  v0 

z-i  -  b 


RT 


1  Traube  \Zeit.f.  anorg.  CAem.,  loc.  cit.\. 


DIETERICPS   LATENT  HEAT  EXPRESSION     105 


b  is,  of  course,   here   assumed   to  be   a   constant,   and   this 

ci         RT 
expression  is  simply  an  equivalent  of  — .     As  already 

VQ  M 

pointed  out,  the  calculated  values  are  in  nearly  all  cases  lower 
than  the  observed,  and  there  is  no  doubt  that  the  discrepancy  is 
due  to  the  variation  of  a  and  b  with  temperature  and  volume. 

For  a  discussion  of  the  internal  pressure  K,  cf.  \V.  C. 
McC.  Lewis,  Trans.  Farad,  Soc.,  7,  part  I.  (1911).  Also  for 
a  comparison  of  Bakker's  expression  with  those  of  Milner,  cf. 
W.  C.  McC.  Lewis,  Zdt.f.phys.  Chem.,  79,  196  (1912). 

Dieter icfs  Expression  for  the  Latent  Heat  of   Vaporisation. 

The  following  empirical  relation  for  the  internal  latent  heat 
of  vaporisation  A;  has  been  found  by  Dieterici  (Ann.  der 
Physik.,  25,  269,  1908;  /#.,  35,  220,  1911)  to  hold  good, 
viz. — 

\  =  CRT  log  ^ 

where  C  is  a  constant,  ?'2  and  v±  are  the  specific  volumes  of 
the  saturated  vapour  and  liquid  respectively,  and  R  refers  to 
one  gram.  The  constancy  of  C  is  shown  by  the  following 
results  based  on  S.  Young's  data  for  iso-pentane — 

Iso-pentane  (fc  —  i87'8°C.j  Tc  =  460-8); pc  =  25,010  mm. 
of  mercury,  vc  =  4/266  c.c. 


Tabs. 

Pressure  of 
saturated 
vapour. 

vi 

V2 

A.  obtained  from 
'   Clapeyron 
equation. 

RT<-; 

C. 

283-0 

390-5 

•5885 

607-5 

78*64  calories 

46-15 

1-704 

293-0 

572-6 

'6141 

424-0 

75-97 

44-76 

1-697 

303-0 

8I5-3 

•6413 

303-0 

73-52 

43-35 

1-696 

323-0 

I533-0 

•7005 

167-6 

68-95 

40-66 

1-696 

343"o 

2653-0 

•7679 

98-9 

64-18 

37-86 

1-695 

363-0 

4296-0 

•8475 

6l'85 

59-49 

34-94 

1-703 

383-0 

6596-0 

'9455 

39-80 

53-78 

31-71 

1-696 

403-0 

9707-0 

2-0720 

26'10 

47-65 

27-99 

1-702 

423-0 

13804-0 

2-2500 

I7T4 

40-18 

23-56 

1-705 

443"o 

19094.0 

2'5550 

IO-7I 

29-53 

17-40 

1-697 

458-0 

23992-0 

3-1830 

6355 

14-17 

8-68 

1-632 

460-8 

250IO-0 

4-266 

4-266 

° 

o 

— 

106        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 


It  will  be  seen  that  the  constant  C  is  really  an  excellent 
constant  for  low  temperatures  up  to  the  neighbourhood  of  the 
critical  point.  That  fhe  above  expression,  invoking  the 
expressions  v±  and  z/2l  should  hold  good  is  very  surprising; 
one  would  have  expected  that  the  "  b  "  correction  should  have 

been  brought  in,  giving  an  expression  containing  log   — 2> 

vi  —  b\ 
but  this  does  not  give  C  a  constant,  unless  we  assume  that 

_!=_?,  which  is  known  to  be  not  the  case.     Not  only  do  we 

^2      /'2 

obtain  a  value  for  C  which  is  independent  of  the  temperature 
for  a  single  substance  such  as  iso-pentane,  it  is  also  found 
that  this  value  for  C  is  approximately  a  general  constant  holding 
for  all  normal — non-associated — substances,  as  the  following 
table  shows.  The  value  for.C  for  each  substance  is  the  mean 
value  obtained  from  a  wide  temperature  range  similar  to  the 
iso-pentane  case.  Dieterici  states  that  the  variation  in  C  for 
each  substance  is  not  more  than  2  per  cent. 


Substance. 

,  C. 

Substance. 

C. 

I*7O7 

Zinc  chloride    . 

1741 

n  -Hexane     .      .      . 

I'7C2 

Ethyl  ether 

1*724. 

n.  -Heptane   . 

1*814 

Methyl  formate 

i  -706 

r8?8 

Ethyl  formate 

1*747 

2  :  3  dimethyl  n.  -Butane 

1725 

Methyl  acetate 

1-784 

di-iso-butylor  2  : 
n.-hexane 

5  dimethyll 

I-8I3 

Propyl  formate 
Ethyl  acetate    .      . 

1-774 
1*812 

Hexamethylene 

1-694 

Methyl  propionate 

1-803 

Benzene   . 
Fluor-benzene 

1-690 
I7II 

Propyl  acetate 
Ethyl  propionate    . 

1*850 
1-837 

Chlor-benzene 

1714 

Methyl  butyrate     . 

1-824 

Brom-benzene 

1-691 

Methyl  iso-butyrate 

i  810 

lodo-benzene 

1-687 

Carbon  dioxide 

1717 

Carbon-tetrachloride 

1-667 

Sulphur  dioxide 

1730 

The  reader  must  be  careful  not  to  confuse  C  with  C0 
(Nernst's  "  Chemical  Constant ")  already  referred  to. 

The  alcohols  and  acetic  acid  do  not  give  a  constant 
independent  of  temperature. 

If  we  were  dealing  with  a  perfect  gas  and  allowed  it  to 


SATURATED   STEAM  107 

expand  from  volume  v±  to  volume  v^  the  work  done  would  be 
RT  log  — *  The  expression  for  the  internal  energy  change 

involved  in  the  vaporisation,  namely  CRT  log  — ,  has  a  formal 

resemblance  to  this,  but  it  must  be  remembered  that  this  latter 
expression  only  holds  for  singular  points  at  each  temperature, 
namely,  the  volumes  of  the  saturated  vapour  and  the  liquid 
respectively.  It  does  not  follow  that  the  energy  difference  will 
be  C  times  the  ideal  work  done  in  general  for  any  volume 
change  in  a  liquid  system.  v±  and  v2  must  only  represent  the 
limits  above  named.  Dieterici  makes  use  of  the  above  con- 
siderations to  show  that  any  equation  of  state,  e.g.  van  der 
Waals',  which  assumes  that  pressure  is  only  due  to  translatory 
motion  of  the  particles,  and  not  in  any  way  connected  with 
internal  motion — perhaps  rotational — is  necessarily  incom- 
plete. (Cf.  Dieterici,  AnnaL  der  Physik.^  35,  229,  1911). 

The  Thermal  Properties  of  Saturated  Water  Vapour  (Steam). 

Suppose  we  consider  i  gram  of  water  consisting  of  (i  —  m) 
grams  of  liquid  and  m  grams  of  steam  in  contact.  The  water 
may  be  considered  either  as  carried  along  with  the  steam, 
producing  "  wet  steam,"  or  as  a  two-layer  system,  liquid  and 
vapour.  (When  m  =  i,  the  system  has  become  entirely  dry 
saturated  steam.)  At  a  given  temperature,  let  the  volume  of 
i  gram  of  water  be  v0,  and  the  volume  of  i  gram  of  steam  z^. 
The  total  volume  of  the  system  V  is  given  by — 

V  =  (i  —  m)vQ  -f-  mv^. 

Now,  suppose  a  small  quantity  of  heat  ^Q  is  added  to  the 
system.  In  consequence  there  will  be  (a)  a  rise  in  tempera- 
ture, (b)  further  evaporation,  for  since  the  vapour  remains 
saturated,  the  higher  the  temperature  the  greater  the  quantity 
(mass)  of  water  is  required  to  saturate  a  given  volume.  Sup- 
pose a  mass  dm  of  water  has  been  turned  into  steam.  Heat 
required  =  l^dm.  Also,  if  at  the  same  time  the  temperature 


i io        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

negative  at  ordinary  temperatures,  it  must  pass  through  a  stage 
at  which  it  is  zero. 

Hence  s2  changes  with  temperature  according  to  the  follow- 
ing scheme  : — 

At  low  temperatures  .     .     .     sz  is  negative. 

At  higher  temperature    .     .     s2  is  zero. 

At  still  higher  temperatures     sz  is  positive. 

At  the  critical  temperature  .     s2  is  negative  and  infinite. 


CHAPTER   IV 

Thermodynamic  criteria  of  chemical  equilibrium  in  general. 


THERMODYNAMIC  CRITERIA  OF  CHEMICAL  EQUILIBRIUM  IN 
GENERAL. 

IN  the  consideration  of  Chemical  Equilibrium  from  the  Kinetic 
standpoint  we  saw  that  equilibrium  could  be  conveniently 
divided  into  two  classes  :— 

(1)  Equilibrium  in  homogeneous  systems,  i.e.  equilibrium 
between  components  in  one  and  the  same  phase. 

(2)  Equilibrium  in  heterogeneous  systems,  i.e.  equilibrium 
between  components  in  different  phases. 

From  the  kinetic  standpoint  we  were  able  to  grasp  the 
important  idea  involved  in  the  term  "  Active  Mass,"  and  we 
saw  how  this  idea  led,  in  the  case  of  homogeneous  systems,  to 
the  generalisation  known  as  the  Law  of  Mass  Action ;  and  in 
the  case  of  heterogeneous  systems  to  the  generalisation  called 
the  Law  of  Partition  or  Distribution. 

From  the  standpoint  of  thermodynamics  we  can  use  the 
same  division  of  equilibrium  into  the  two  classes,  homogeneous 
and  heterogeneous.  As  we  shall  see,  we  can  arrive  at  the  Law 
of  Mass  Action  (without  introducing  the  molecular  hypothesis) 
as  the  criterion  for  homogeneous  equilibrium  under  certain  con- 
ditions ;  and,  further,  with  the  help  of  thermodynamical  reason- 
ing in  the  case  of  heterogeneous  equilibrium,  we  arrive  at  a 
much  wider  principle,  in  addition  to  the  Distribution  Law, 
namely,  the  so-called  Phase  Rule.  We  shall  study  these  in 
turn  later  on.  For  the  present,  however,  it  is  necessary  to  con- 
sider the  general  problem  of  equilibrium. 

First  of  all  a  few  typical  instances  of  physical  and  chemical 
equilibrium  may  be  mentioned.  The  simplest  type  of 


H2        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

equilibrium  is  that  represented  by  a  liquid  in  contact  with  satu- 
rated vapour  in  an  enclosed  vessel.  The  system  will  remain  in 
an  unchanged  state  for*  infinite  time,  the  "reaction"  in  this  case, 
which  has  reached  an  equilibrium,  is  the  transfer  of  molecules 
from  the  liquid  to  the  vapour  and  vice  versa.  This  happens  to 
be  an  instance  of  heterogeneous  equilibrium.  As  an  illustration 
of  homogeneous  equilibrium  we  can  take  the  case  of  gaseous 
hydriodic  acid  in  a  state  of  partial  dissociation  into  hydrogen 
(H2)  and  iodine  (I2),  or  gaseous  water  in  equilibrium  with 
hydrogen  and  oxygen ;  or  we  can  take  the  case  of  the  equili- 
brium reached  when  acetic  acid  and  ethyl  alcohol  are  mixed 
together  producing  some  ethyl  acetate  and  water,  all  four  sub- 
stances being  present  together  at  certain  concentration  values, 
in  the  equilibrium  state ;  or  the  equilibrium  which  is  reached 
when  water  is  added  to  sulphuric  acid,  giving  rise  to  some 
addition  compound,  thus — 

(H2S04)(H20);,  J;;H20  +  H2SO4 

or,  finally,  we  can  take  the  case  of  an  electrolyte  dissociating  in 
an  ionising  solvent,  equilibrium  being  established  (practically 
instantaneously)  between  the  unionised  molecules  of  the  solute 
and  the  ions.  When  we  come  to  consider  any  system  in  which 
a  reaction  may  occur  such  a  system  will  always  tend  to  reach 
an  equilibrium  state.  Some  systems  never  do  reach  a  true 
equilibrium  state.  They  are  under  all  conditions  meta-stable. 
ThiSj  however,  is  due  to  the  slowness  of  the  rate  at  which  they 
are  progressing  towards  the  equilibrium.  Such  systems  give 
rise  to  an  apparent  equilibrium  state  or  false  equilibrium.  These, 
however,  need  not  concern  us  further  here,  for  thermodynamics 
has  nothing  quantitative  to  say  to  such  cases.  We  are  here 
only  considering  reactions  of  any  kind  whatsoever  which  do 
within  a  measurable  time  reach  the  permanent  state  of  equili- 
brium. The  time  criterion  of  equilibrium  is,  of  course,  that 
the  system  remains  unchanged  for  infinite  time.  This,  although 
true,  cannot  help  us  from  the  theoretical  standpoint,  for,  of 
course,  we  cannot  observe  any  system  for  infinite  time.  It  has, 
however,  a  very  practical  use  as  an  approximation.  We  are  at 
present  trying  to  find  out  thermodynamic  criteria  which  must 


CHEMICAL  EQUILIBRIA  113 

be  satisfied  when  a  system  has  come  into  the  true  equilibrium 
state.  Let  us  pause  to  consider  the  conditioning  factors,  or,  as 
they  are  called,  the  parameters  or  "  variables,"  by  altering  any  of 
which  a  system  may  in  general  be  altered,  i.e.  the  factors  which 
determine  whether  a  reaction  will  proceed  or  not.  These 
factors  may  be  large  in  number — temperature,  pressure,  con- 
centration, electric  state,  capillary  state,  magnetic  state,  etc. 
Take  a  system  depending  on  the  first  three  of  these  factors,  as 
is  frequently  the  case,  namely,  the  temperature,  pressure,  and 
concentration  of  the  reacting  substances.  Any  equilibrium  may 
be  regarded  as  depending  on  these  factors.  These  three  factors 
are,  however,  not  all  independent  of  one  another.  Suppose  we 
take  the  simplest  case  of  a  gas  enclosed  in  a  vessel.  We  take 
a  certain  mass  of  gas  and  the  vessel  has  a  certain  volume. 
That  is,  we  chose  the  system  at  a  certain  arbitrary  concentra- 
tion. This  system  can  exist  as  such,  i.e.  as  a  gas  at  various 
temperatures  and  various  pressures.  Experience  shows,  how- 
ever, that  if  we  arbitrarily  select  a  definite  temperature  (having 
previously  selected  a  certain  mass  in  a  certain  volume,  i.e.  a 
certain  concentration)  the  pressure  will  adjust  itself  to  a  certain 
value  characteristic  of  the  equilibrium  state.  In  the  case  of 
liquid  water  in  contact  with  water  vapour  in  an  enclosed  vessel, 
if  we  arbitrarily  choose  a  certain  temperature  then  the  other 
parameters  will  be  fixed  when  the  system  is  in  equilibrium,  i.e. 
the  pressure  is  the  pressure  of  saturated  vapour  at  this  tempera- 
ture, and  the  concentration  or  density  terms  of  the  liquid  and 
vapour  phases  have  certain  values.  Such  considerations  as 
these,  namely,  the  number  of  factors  which  define  equilibrium 
and  their  interdependence,  will  be  taken  up  in  dealing  with  the 
Phase  Rule.  For  the  present  we  shall  proceed  along  some- 
what different  lines. 

An  equilibrium  state  is  defined  as,  that  state  reached  when 
some  thermodynamic  quantity  (called  a  "  thermodynamic 
potential ")  has  reached  its  minimal  value,  i.e.  when  it  can  no 
longer  decrease.  We  have  to  find  out  what  thermodynamical 
potential  we  must  regard  as  justifiable  to  apply  to  the  various 
cases  which  occur.  Free  energy  is  one  of  the  thermodynamical 
potentials,  and  as  we  are  familiar  with  it  (Chap.  I.,  Part  II.), 
T.P.C. — ii.  i 


H2        A    SYSTEM   OF  PHYSICAL    CHEMISTRY 

equilibrium  is  that  represented  by  a  liquid  in  contact  with  satu- 
rated vapour  in  an  enclosed  vessel.  The  system  will  remain  in 
an  unchanged  state  for*  infinite  time,  the  "  reaction  "  in  this  case, 
which  has  reached  an  equilibrium,  is  the  transfer  of  molecules 
from  the  liquid  to  the  vapour  and  vice  versa.  This  happens  to 
be  an  instance  of  heterogeneous  equilibrium.  As  an  illustration 
of  homogeneous  equilibrium  we  can  take  the  case  of  gaseous 
hydriodic  acid  in  a  state  of  partial  dissociation  into  hydrogen 
(H2)  and  iodine  (I2),  or  gaseous  water  in  equilibrium  with 
hydrogen  and  oxygen ;  or  we  can  take  the  case  of  the  equili- 
brium reached  when  acetic  acid  and  ethyl  alcohol  are  mixed 
together  producing  some  ethyl  acetate  and  water,  all  four  sub- 
stances being  present  together  at  certain  concentration  values, 
in  the  equilibrium  state ;  or  the  equilibrium  which  is  reached 
when  water  is  added  to  sulphuric  acid,  giving  rise  to  some 
addition  compound,  thus — 

(H2S04)(H20),  J>H20  +  H2S04 

or,  finally,  we  can  take  the  case  of  an  electrolyte  dissociating  in 
an  ionising  solvent,  equilibrium  being  established  (practically 
instantaneously)  between  the  unionised  molecules  of  the  solute 
and  the  ions.  When  we  come  to  consider  any  system  in  which 
a  reaction  may  occur  such  a  system  will  always  tend  to  reach 
an  equilibrium  state.  Some  systems  never  do  reach  a  true 
equilibrium  state.  They  are  under  all  conditions  meta-stable. 
ThiSj  however,  is  due  to  the  slowness  of  the  rate  at  which  they 
are  progressing  towards  the  equilibrium.  Such  systems  give 
rise  to  an  apparent  equilibrium  state  or  false  equilibrium.  These, 
however,  need  not  concern  us  further  here,  for  thermodynamics 
has  nothing  quantitative  to  say  to  such  cases.  We  are  here 
only  considering  reactions  of  any  kind  whatsoever  which  do 
within  a  measurable  time  reach  the  permanent  state  of  equili- 
brium. The  time  criterion  of  equilibrium  is,  of  course,  that 
the  system  remains  unchanged  for  infinite  time.  This,  although 
true,  cannot  help  us  from  the  theoretical  standpoint,  for,  of 
course,  we  cannot  observe  any  system  for  infinite  time.  It  has, 
however,  a  very  practical  use  as  an  approximation,  We  are  at 
present  trying  to  find  out  thermodynamic  criteria  which  must 


CHEMICAL   EQUILIBRIA  113 

be  satisfied  when  a  system  has  come  into  the  true  equilibrium 
state.  Let  us  pause  to  consider  the  conditioning  factors,  or,  as 
they  are  called,  the  parameters  or  "  variables,"  by  altering  any  of 
which  a  system  may  in  general  be  altered,  i.e.  the  factors  which 
determine  whether  a  reaction  will  proceed  or  not.  These 
factors  may  be  large  in  number — temperature,  pressure,  con- 
centration, electric  state,  capillary  state,  magnetic  state,  etc. 
Take  a  system  depending  on  the  first  three  of  these  factors,  as 
is  frequently  the  case,  namely,  the  temperature,  pressure,  and 
concentration  of  the  reacting  substances.  Any  equilibrium  may 
be  regarded  as  depending  on  these  factors.  These  three  factors 
are,  however,  not  all  independent  of  one  another.  Suppose  we 
take  the  simplest  case  of  a  gas  enclosed  in  a  vessel.  We  take 
a  certain  mass  of  gas  and  the  vessel  has  a  certain  volume. 
That  is,  we  chose  the  system  at  a  certain  arbitrary  concentra- 
tion. This  system  can  exist  as  such,  i.e.  as  a  gas  at  various 
temperatures  and  various  pressures.  Experience  shows,  how- 
ever, that  if  we  arbitrarily  select  a  definite  temperature  (having 
previously  selected  a  certain  mass  in  a  certain  volume,  i.e.  a 
certain  concentration)  the  pressure  will  adjust  itself  to  a  certain 
value  characteristic  of  the  equilibrium  state.  In  the  case  of 
liquid  water  in  contact  with  water  vapour  in  an  enclosed  vessel, 
if  we  arbitrarily  choose  a  certain  temperature  then  the  other 
parameters  will  be  fixed  when  the  system  is  in  equilibrium,  i.e. 
the  pressure  is  the  pressure  of  saturated  vapour  at  this  tempera- 
ture, and  the  concentration  or  density  terms  of  the  liquid  and 
vapour  phases  have  certain  values.  Such  considerations  as 
these,  namely,  the  number  of  factors  which  define  equilibrium 
and  their  interdependence,  will  be  taken  up  in  dealing  with  the 
Phase  Rule.  For  the  present  we  shall  proceed  along  some- 
what different  lines. 

An  equilibrium  state  is  defined  as,  that  state  reached  when 
some  thermodynamic  quantity  (called  a  "  thermodynamic 
potential ")  has  reached  its  minimal  value,  i.e.  when  it  can  no 
longer  decrease.  We  have  to  find  out  what  thermodynamical 
potential  we  must  regard  as  justifiable  to  apply  to  the  various 
cases  which  occur.  Free  energy  is  one  of  the  thermodynamical 
potentials,  and  as  we  are  familiar  with  it  (Chap.  I.,  Part  II.), 
T.P.C. — ii.  i 


H4        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

we  can  commence  by  saying  that  the  free  energy  of  any  system 
tends  to  decrease.  Notice  particularly,  however,  that  we  do 
not  say  equilibrium  in  general  is  reached  when  the  free  energy 
of  the  system  is  a  minimum.  This  latter  statement  as  a  general 
law  would  be  quite  untrue,  as  will  be  shown  later. 

We  must,  in  fact,  first  postulate  two  things  (i)  that  the  tem- 
perature is  supposed  to  be  maintained  constant,  and  (2)  that 
the  reaction  under  consideration  can  proceed  without  a  volume 
change.  Now  scarcely  any  reaction  takes  place  without  a 
volume  change,  but  in  condensed  systems,  as  van  't  HofT 
called  them,  that  is  in  systems  either  liquid  or  solid  (homo- 
geneous or  heterogeneous)  we  can  imagine  the  reaction  to 
occur  practically  without  a  volume  change. 

The  criterion  of  equilibrium  being  reached  in  such  a  reaction, 
i.e.  in  one  in  which  the  temperature  and  volume  are  maintained 
constant,  is  simply  that  at  the  equilibrium  point  the  Free  Energy 
is  a  minimum,  and  therefore  if  we  consider  tlie  system  when 
equilibrium  is  reached  and  imagine  the  reaction  to  go  to  a  small 
extent,  namely,  the  transformation  of  §n  molecules  from  one  side 
of  the  equilibrium  to  the  other,  then  the  work  done  or  free  energy 
change  is  zero.  This  is  written  algebraically  (8f)  =  o.1 

We  can  likewise  denote  this  by  (SA)TV  =  o,  where  A 
denotes  external  work.2  Practically  all  the  external  isothermal 
reversible  mechanical  work  processes  with  which  we  have  to 
deal  consist  simply  of  the  three-stage  work  process  already 
discussed  at  some  length.  Although  this  process  involves  a 
vaporisation  step,  which  is  one  entailing  a  large  volume  change, 
the  final  step  of  the  process  entails  a  condensation,  and  it  is 

1  Those  familiar  with  the  principle  of  maxima  and  minima  in  the 
differential  calculus  will  see  that  the  above  equation  represents  mathe- 
matically a  maximum  point  on  a  continuous  curve  as  well  as  a  minimum. 

More  strictly  we  should  write  for  the  equilibrium  criterion  (S/)TV  =  o  ; 
but  the  physical  or  chemical  significance  of  the]; equality  relation,  viz. 
(§/)TV  =  o,  will  cause  no  confusion. 

-  This  is  known  as  the  chemical  application  of  the  Principle  of  Virtual 
Work.  For  an  account  of  Virtual  Work  in  Mechanics,  for  example,  see 
A.  W.  Porter's  book,  Intermediate  Mechanics. 


CRITERIA    FOR  EQUILIBRIUM  115 

therefore  possible  to  conceive  of  this  three-stage  process  as 
being  carried  out  practically  without  a  volume  change.  Let  us 
think  of  the  system  ice  and  water  not  in  equilibrium  at  some 
temperature  below  o°  C.,  but  under  the  pressure  of  their  own 
vapours.  There  is  a  tendency  of  the  water  to  solidify,  this 
being  a  chemical  reaction  as  much  as  any  other  reaction  is. 
We  can  imagine  i  mole  of  the  substance  (water)  transferred 
by  the  three-stage  distillation  process  from  the  liquid  water  to 

the  ice  (at  constant  temperature),  the  work  A  amounting  to 
i 
vdp)  which  will  be  a  positive  work  output  (and  therefore  a 

real  decrease  in  free  energy)  if  the  vapour  pressure  over  the 
water  (I)  is  greater  than  that  over  ihe  ice  (II).  Experience 
has  shown  that  below  o°  the  vapour  pressure  over  the  liquid 
(super-cooled  and  meta-stable  water)  is  greater  than  that  over 
ice,  so  there  is  a  real  decrease  in  f  in  the  system  when  the 
liquid  water  becomes  solid.  Now  the  actual  change  in 
volume  in  this  reaction  is  exceedingly  small,  it  being  simply 
the  difference  of  the  molecular  volumes  of  ice  and  water 
respectively,  for  the  total  change  in  volume  in  any  reaction  is 
evidently  the  final  volume  reached  minus  the  initial  volume, 
no  matter  what  volume  changes  may  have  occurred  en  route. 
We  can  thus  imagine  that  the  transformation  of  liquid  water 
to  ice  can  take  place  at  constant  temperature  and  volume. 
Ice  and  water  are  in  equilibrium  by  definition  when  (8f)  or 

(SA)  =  o.  Suppose  they  are  in  equilibrium  (experiment 
shows  that  one  such  equilibrium  state  will  be  reached  when 
T  is  2734-o'oo7°j  and  the  pressure  is  the  vapour  pressure 
of  the  substances).  If  we  imagine  a  small  quantity,  8n  moles, 
of  water  transferred,  say,  from  ice  to  liquid  water  under 

these   conditions,   then  according  to  definition  we  must  set 

ri 
Work  =  zero.     Now  the  work  is  evidently  given  by  8n  I     vdp> 

J  n 

and  in  order  that  this  may  be  zero  it  is  evident  that  the 
pressure  of  the  ice  and  of  the  water  (I  and  II)  must  be  identical 
so  as  to  make  dp  =  o.  This  is  an  experimental  fact, 
viz.,  ice  and  water  are  in  equilibrium  when  the  vapour 


ii6        A    SYSTEM   OF  PHYSICAL    CHEMISTRY 

pressures  are  identical,  and  this  occurs  at  +  0-007°  C.  when 
water  is  under  its  o\*n  vapour  pressure.  Equilibrium  is  also 
reached  at  o°  C.,  when  these  pressures  are  modified  by  the 
presence  of  the  atmosphere  though  again  the  vapour  pressures 
of  both  (ice  and  water)  are  identical.  [Note :  effect  of 
an  inert  gas  on  the  vapour  pressure.  From  the  molecular 
standpoint  we  see  that  the  presence  of  the  inert  gas  on  the 
surface  of  the  liquid  will  to  a  certain  extent  compress  the 
liquid,  and  thus  cause  an  increase  in  the  number  of  liquid 
molecules  distributed  over  each  unit  area  of  liquid  surface. 
The  chance  of  a  molecule  leaving  the  liquid  is  thereby 
increased,  and  hence  the  vapour  pressure  of  the  liquid  is 
increased,  at  least  this  is  the  usual  explanation.  In  the  case 
of  ice  and  water  at  o°  C.,  and  under  the  pressure  of  the 
atmosphere,  the  vapour  pressure  over  each  phase  is  increased, 
but  to  the  same  extent  so  that  the  vapour  pressures  are  still 
identical] 

Or  take  again  the  reaction  involved  in  the  transformation 
of  rhombic  into  monoclinic  sulphur,  or  grey  tin  into  white  tin. 
By  following  exactly  the  same  process  of  reasoning  and 
carrying  out  an  imaginary  work  process  at  constant  tempera- 
ture and  volume,  we  find  that  the  allotropic  forms  are  in 
equilibrium  when  the  vapour  pressures  are  identical.  Again 
the  reaction  between  liquid  water  and  sulphuric  acid  may  also 
be  reached  at  constant  temperature,  and  practically  constant 
volume ;  for  any  change  in  volume  in  the  whole,  is  the 
difference  of  molecular  volume  of  the  water  in  the  pure  state, 
and  its  volume  in  sulphuric  acid  solution — a  quantity  which  is 
not  large.  The  principle  that  equilibrium  is  reached  when 
(8f)Ty=  o,  or  (8A)TV=  o  will  apply,  for  here  again  the 

three-stage  distillation  process  must  be  "zero  at  the  equilibrium 
point,  which  requires  that  the  pressures  over  the  water  and 
strong  sulphuric  acid  solution  shall  be  equal.  But  this  we 
know  by  experience  can  never  be  reached  as  long  as  any  liquid 
water  remains;  for  the  vapour  pressure  over  pure  water  is 
always  greater  than  the  vapour  pressure  over  solutions  con- 
taining water  (at  the  same  temperature)  and  hence  the  criterion 


• 


CRITERIA    FOR  EQUILIBRIUM  117 

leads  us  to  expect  that  equilibrium  can  only  be  reached  when 
all  the  liquid  water  disappears  and  the  vapour  pressure  falls 
to  that  over  the  sulphuric  acid  water  solution.  This  is  what 

;;tually  happens. 
Further,  the  criterion  that  equilibrium  is  reached  when 
A)TV  =  o,  applies  to  reactions  in  solutions.  Take  the 
uplest  case  of  diffusion  of  a  solute  from  a  region  of  high 
concentration  to  one  of  low.  Equilibrium  is  reached  when 
the  work  entailed  by  a  small  virtual  change  in  the  system 
can  be  equated  to  zero.  Suppose  we  have  a  solute  at  con- 
centrations Ci  and  cz  and  osmotic  pressures  Px  and  P2 
in  the  same  vessel.  By  means  of  osmotic  membranes  (the 
whole  vessel  being  supposed  to  be  surrounded  by  solvent  of 
infinite  extent)  we  can  imagine  8n  moles  transferred  from 

fpi 

PI  to  P2  the  work  being  8n  I    vdP,  which  will  be  zero  when 

P2 

P!  =  P2  or  e±  =  c2 ;  that  is  equilibrium  is  reached  when  the 
concentration  is  the  same  throughout  the  vessel ;  this  we  know 
is  an  experimental  fact. 

The  criterion  we  have  been  considering  breaks  down, 
however,  when  the  reaction  cannot  be  conceived  of  as  occur- 
ring with  even  approximately  constant  volume.  Thus,  take  the 
reaction  such  as  vaporisation  or  sublimation,  i.e.  change  from 
a  condensed  to  a  gaseous  system  not  followed  by  condensa- 
tion. This  can  be  carried  out  at  constant  temperature,  but  not 
at  constant  volume,  though  it  can  be  done  at  constant  pressure. 
We  know  from  experience  that  equilibrium  is  reached  when 
the  vapour  has  reached  saturation  pressure,  but  when  we  con- 
sider such  a  system  in  the  equilibrium  state,  and  suppose  a 
quantity,  8n  moles,  say,  vaporised,  the  work  is  not  zero ;  the 
work  is  the  pressure  into  the  change  in  volume.  We  must  infer, 
therefore,  since  work  is  done  by  the  system  in  passing  from 
liquid  to  vapour,  that  the  free  energy,  say,  of  i  gram  or 
i  gram-mole  of  liquid  is  greater  than  the  free  energy  of 
i  gram  or  i  gram-mole  of  saturated  vapour  at  one  and  the 
same  temperature,  even  though  the  liquid  and  vapour  are  in 
equilibrium.  Obviously,  if  the  free  energy  of  the  one  phase  is 


ii8        A   SYSTEM  OF  PHYSICAL    CHEMISTRY 

greater  than  the  other,  the  free  energy  of  the  system  is  not 
a  minimum.  One  must  be  careful,  therefore,  not  to  say  that  in 
general  at  a  given  temr5erature  the  equilibrium  is  reached  when 
the  free  energy  of  the  system  is  a  minimum ;  for  if  this  were 
so,  then  in  the  liquid-vapour  system  just  considered  one  would 
expect  the  whole  system  to  become  vapour,  since  the  vapour 
has  less  free  energy  \  this  is,  however,  contrary  to  experience. 
We  must  bring  in  the  restriction  that  the  process  can  be 
carried  out  at  constant  volume  (or  approximately  so)  as  well  as 
at  constant  temperature.  In  such  a  case  the  equilibrium  is 
reached  when  the  free  energy  is  a  minimum.  It  may  be 
pointed  out,  however,  that  in  the  liquid-vapour  equilibrium, 
although  the  free  energies  of  a  given  mass  of  liquid  and  vapour 
respectively  differ,  there  is  a  quantity  called  the  "  thermodyna- 
mical  potential  at  constant  temperature  and  pressure,"  which  is 
denoted  by  the  symbol  0,  and  this  has  the  same  numerical 
value  for  i  gram  of  saturated  vapour  as  it  has  for  the  same 
mass  of  liquid  at  one  and  the  same  temperature.  In  reactions 
which  occur  with  change  of  volume,  but  at  constant  tempera- 
ture and  pressure,  equilibrium  is  reached  when  the  0  of  the 
system  is  a  minimum.  To  appreciate  the  significance  of  the 
thermodynamical  quantity  0  we  must,  however,  present  the 
First  and  Second  Laws  in  a  more  analytical  way  than  we  have 
done  in  the  chapter  in  the  "  Elementary  Treatment."  For 
those  who  have  followed  the  chapter  on  the  "  more  advanced 
treatment"  (Chap.  II.)  a  short  discussion  of  the  0  function  is 
added  to  the  present  section.  Before  leaving  the  consideration 
of  reactions  which  can  be  carried  out  at  constant  volume  (ap- 
proximately) and  at  constant  temperature,  it  is  essential  to  speak 
of  those  which  are  capable  of  losing  free  energy  in  the  form  of 
electrical  energy,  the  reacting  substances  being  set  up  in  the 
form  of  a  cell.  The  importance  of  this  is  due  to  the  fact  that 
where  a  cell  is  capable  of  being  set  up  at  all,  the  measurement 
of  the  electromotive  force,  which  can  be  always  carried  out 
with  great  accuracy,  gives  us  quantitative  information  of 
exactly  how  far  the  system  is  from  the  equilibrium  point.  We 
shall  have  occasion  later  on  to  make  great  use  of  this  in  the 
measurement  of  affinity. 


THE   DANIELL    CELL  119 

Consider  the  reaction  which  takes  place  when  a  piece  of 
metallic  zinc  is  placed  in  a  solution  of  copper  sulphate.  Copper 
is  precipitated  in  the  metallic  form,  and  zinc  sulphate  solution 
is  produced.  The  reaction  is  written  in  the  form — 

Zn  +  CuSO4aq  =  Cu  +  ZnSO4aq. 
In  terms  of  ions  we  can  represent  it  by — 

Zn  +  Cu  ++  =  Zn  ++  +  Cu. 

If  this  reaction  simply  takes  place  in  a  test-tube,  heat  is 
evolved,  but  no  work  is  done.  If  the  tube  be  immersed  in  a 
thermostat,  the  reaction  can  be  made  to  go  at  practically  con- 
stant temperature  and  volume.  To  get  work  from  this  reaction, 
and  especially  to  get  the  maximum  work  from  it,  it  is  necessary 
to  make  it  proceed  reversibly  by  means  of  a  suitable  thermo- 
dynamic  arrangement.  This  arrangement  is  very  nearly  realised 
by  the  setting  up  of  the  system  in  the  form  of  an  electrolytic  cell, 
i.e.  the  Daniell  cell,  having  an  electrode  of  copper  immersed  in 
copper  sulphate  solution,  and  an  electrode  of  zinc  in  a  zinc 
sulphate  solution.  Under  these  conditions,  electrical  energy 
(electromotive  force  X  quantity  of  current)  can  be  obtained 
from  the  cell,  i.e.  from  the  reaction.  The  cell  must  be  made 
to  work  reversibly  if  there  is  to  be  production  of  maximum 
work.  Suppose  the  cell  possesses  an  e.m.f.  of  E  volts 
(measured  electrostatically,  or,  what  is  the  same  thing,  with  an 
extremely  large  resistance  in  the  outer  circuit),  and  is  made  to 
drive  something  which  possesses  a  back  e.m.f.  of  E  —  */E 
volts,  then  the  cell  is  just  able  to  do  the  work  required  of  it 
and  nothing  more.  Note,  that  we  must  always  reckon  energy 
per  certain  quantity  of  substance  (say,  copper)  transformed  in 
the  sense  of  the  reaction  studied.  It  is  usual,  especially  in 
electrical  measurements,  to  consider  the  gram  equivalent  as 
the  unit  of  mass  transferred.  Now,  the  amount  of  electricity 
associated  with  i  gram  equivalent  is  i  faraday,  so  that  if 
the  e.m.f.  of  the  cell  is  E,  and  one  gram  equivalent  is  trans- 
formed, the  electrical  energy  output  is— 

E  X  i  faraday  =  E  X  i  =  E 


120        A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

That  is,  the  numerical  value  in  volts  of  the  e.m.f.  itself  gives  us 
the  electrical  energy  (in  volt-faradays)  produced  per  gram 
equivalent,  and  from  9  measurement  of  the  e.m.f.  of  the  cell  in 
the  manner  indicated,  we  can  then  say  that  the  maximum  work, 
electrical  in  form,  which  the  reaction  is  capable  of  pro- 
ducing at  the  given  temperature  and  practically  without  volume 
change,  is  E  energy  units  per  equivalent  of  one  of  the  sub- 
stances transformed.  Notice  that  measuring  the  e.m.f.  in  this 
way  gives  us  the  same  result  as  if  we  had  set  up  some  machine 
having  an  opposing  e.m.f.  of  E  —  dE  volts,  and  allowed  the 
cell  to  drive  this  infinitely  slowly.  This  is  quite  analogous  to 
the  idea  of  a  piston  moving  and  doing  mechanical  work  with 
the  pressure  on  the  two  sides  only  differing  by  dp.  In  fact, 
electromotive  force  can  be  regarded  as  "  electrical  pressure  " 
(though  one  cannot  give  it  the  dimensions  of  pressure,  namely, 

,  ,j,2j.  Further,  the  reversibility  of  the  Daniell  cell  is  com- 
pleted by  the  reaction  being  of  such  a  kind  that  an  external 
e.m.f.  of  magnitude  E  -j-  </E  applied  in  the  opposite  sense  to 
that  of  the  cell  is  capable  of  causing  the  chemical  reaction 
also  to  reverse  itself  without  the  production  of  substances  (such 
as  O2  and  H2)  not  previously  present  in  the  system.  In  brief, 
the  electrodes  must  be  reversible  in  the  electro-chemical  sense 
(see  later  Nernst's  Osmotic  Theory  of  Electromotive  Force)) 
that  is,  the  cell  must  not  be  polarisable ;  and  the  direct  and 
reverse  chemical  processes  should  be  capable  of  an  infinite 
number  of  alternate  repetitions  without  any  permanent  change 
in  the  composition  of  the  cell.  Such  a  cell,  working  at 
constant  temperature  and  producing  electrical  energy  infinitely 
slowly,  is  the  nearest  practical  approach  to  the  ideal  conception 
of  a  thermodynamic  reversible  isothermal  process. 

If,  instead  of  working  the  cell  so  as  to  give  maximum  work, 
we  go  to  the  other  extreme  and  short  circuit  the  electrodes, 
thereby  allowing  the  reaction  to  go  quickly,  the  cell  runs  down, 
the  e.m.f.  finally  being  zero,  i.e.  the  cell  is  no  longer  capable 
of  doing  any  work.  The  equilibrium  point  of  the  reaction— 


ALLOTROPIC   CHANGE  12 1 

is  then  reached.  If  now  we  analyse  the  solutions  in  the  cell, 
we  find  that  practically  1  all  the  copper  is  deposited,  and  the 
liquid  is  a  solution  of  zinc  sulphate,  containing  a  certain 
amount  of  Zn  +  +.  If  we  had  simply  allowed  the  reaction  to 
occur  in  a  test-tube  (starting  with  the  same  amount  of  copper 
sulphate  solution  and  adding  metallic  zinc),  we  would  find  on 
analysis  that  the  amount  of  zinc  in  solution  is  the  same  in 
both  cases.  This  is  experimental  evidence  that  the  equili- 
brium is  reached  when  (SA)Tv  =  o. 

The  consideration  of  the  Daniell  cell  will  have  made  clear 
the  distinction  to  be  drawn,  as  far  as  work  production  is  con- 
cerned, between  the  spontaneous  or  irreversible  mode  of 
carrying  out  a  reaction  and  the  reversible  mode  (in  the  thermo- 
dynamical  sense)  of  carrying  out  the  same  reaction,  even  though 
the  end  point  reached  is  identical  in  both  cases. 


APPLICATIONS  OF  THE  FOREGOING  PRINCIPLES  TO  THE 
TRANSFORMATION  OF  ALLOTROPIC  SOLIDS. 

The  systems  we  are  about  to  consider  are  those  consisting 
of  allotropic  solids,  i.e.  different  crystalline  modifications  of  the 
same  chemical  substance  which  possess  the  property  of  trans- 
formation the  one  into  the  other,  the  direction  of  the  trans- 
formation depending  on  the  temperature.  Such  systems  are 
characterised  by  the  existence  of  a  transition  point  or  transition 
temperature  at  which  the  different  crystalline  forms  of  the  one 
substance  are  in  equilibrium  with  one  another.  (The  reason 
why  such  substances  exhibit  a  transition  temperature  will  be 
given  when  we  come  to  apply  the  Phase  Rule.  For  the  present 
we  are  simply  dealing  with  the  phenomenon  as  an  experimental 
fact.)  At  temperatures  below  the  transition  point  one  crystal- 
line form  is  unstable,  /.<?.  tends  to  pass  entirely  into  the  other. 
Above  the  transition  point  the  second  phase  is  now  unstable 
and  will  pass  entirely  into  the  first.  All  these  changes  occur- 
ring at  a  given  temperature  involve  practically  no  change  in 

1  The  infinitely  small  quantity  of  copper  remaining  in  solution  in  this 
particular  reaction  is  far  below  the  limits  of  the  most  delicate  analysis. 


122        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

volume  and  we  can  therefore  apply  the  criterion  that  at  equili- 

/* 
brium  (3A)TV  =  o,  orW   vdp  =  o,  where  the  integral  stands  for 

the  familiar  three-stage  distillation  work  process.  We  shall 
apply  this  criterion  to  prove  the  following  statements,  each  of 
which  has  been  demonstrated  by  experiment. 

(1)  At  the  transition  temperature  the  vapour  pressures  of 
each  modification  is  the  same,  and  the  solubility  in  a  given 
solvent  is  the  same. 

(2)  At  any  given  temperature  the  stable  modification  has  a 
smaller  vapour  pressure  than  the  unstable,  the  temperature  here 
considered  is  necessarily  not  the  transition  point. 

(3)  At  any  given   temperature  other  than  the  transition 
temperature  the   stable  modification  has  a  smaller   solubility 
than  the  unstable  in  a  given  solvent. 

(4)  The   stable  modification   has  a  higher  melting   point 
than  the  unstable  form. 

i.  At  the  Transition  Point  the  Vapour  Pressures  of  both  Modi- 
fications are  Identical  as  are  also  their  Solubilities  in  a  Given 
Solvent. 

At  the  transition  point  by  definition  the  two  modifications 
are  in  equilibrium.  If  now  a  mass  8n  be  transferred  isother- 
mally  and  reversibly  (by  means  of  the  three-stage  distillation 
process)  from  modification  I  to  modification  II,  the  work  must 

/^i 
be  zero,  i.e.  (SATV  =  o,  which  in  this  case  becomes  8n  I    vdp  =  o). 

J  n 

Since  8n  and  v  are  positive  terms,  it  follows  that  in  order  to 
make  the  work  expression  zero,  dp  must  be  zero ;  in  other  words, 
that  the  vapour  pressures  over  I  and  II  must  be  identical,  which 
was  to  be  shown.  Further,  we  could  imagine  the  same  virtual 
work  process  carried  out  not  by  distillation  but  by  means  of 
osmotic  work  involving  three  stages.  The  two  modifications  are 
supposed  to  be  in  contact  with  a  saturated  solution  of  each  in  two 
vessels.  The  same  solvent  is  of  course  employed  and  the  whole 
system  may  be  imagined  as  immersed  in  a  bath  of  the  solvent. 
By  means  of  a  piston  permeable  to  solvent  but  impermeable  to 


ALLOTROPIC   CHANGE  123 

solute,  the  quantity  8n  moles  of  modification  I  is  dissolved  by 
drawing  out  the  piston,  the  solution  containing  this  quantity  at 
osmotic  pressure  Pz  being  now  expanded,  i.e.  diluted  to  osmotic 
pressure  Pn,  and  then  compressed  into  the  saturated  solution  of 
modification  II  at  osmotic  pressure  Pu.  The  total  work  is 


£ 

J 


vdP,  and  this  must  be  zero,  which  means  that  Px  must 
pn 

be  identical  with  Pn.  If  the  substance  in  solution  obeys  the 
gas  laws,  the  identity  of  osmotic  pressure  means  identity  of 
concentration,  and  since  solid  was  present  in  each  case  this 
means  identity  of  solubility  as  was  to  be  shown. 

2.  At  any  other  Temperature  the  Stable  Modification  has  a  Lower 
Vapour  Pressure  than  the  Unstable  at  the  same  Tempera- 
ture. 

We  commence  by  taking  it  as  true  that  the  meta-stable 
always  tends  to  pass  into  the  stable.  This  -tendency  only 
exists  in  virtue  of  the  system  being  ready  to  do  positive  work. 
Let  us  imagine  the  transfer  of  6//  moles  from  the  unstable 
modification  to  the  stable  modification  ;/0  refers  to  the  vapour 
pressure  of  the  unstable  modification,  and  p±  to  that  of  the 


stable.     The   three-stage  work  done   is,  8n        vdp.      If  the 

J  /i 
vapours  obey  the  gas  laws,  we  can  write  — 

8A=8«RT  log  ^. 

P\ 

Now  since  the  transfer  has  taken  place  from  the  unstable  to 
the  stable  modification,  this  must  mean  that  positive  work  SA  is 

done  by  the  system.     Since  SAis  positive,  log  --  must  be  posi- 

P\ 

tive.  That  is  /0  must  be  greater  than  p±.  But  /0  is  the  vapour 
pressure  of  the  unstable  phase.  Hence  the  vapour  pressure  of 
the  unstable  modification  is  greater  than  that  of  the  stable.  (It 
will  be  observed  that  this  would  also  hold  even  if  the  vapours 

f&9 

do  not  obey  the  gas  laws,  for  in  order  that         vaP  mav  De 

J  P\ 


124        A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

positive  the  upper  limit  pQ  must  be  greater  than  the  lower 
limit /j.) 


3.   The  Unstable  Modification  has  the  Greater  Solubility. 

One  pursues  exactly  the  same  train  of  reasoning  as  in  Case 
2,  using,  however,  osmotic  pressure  instead  of  vapour  pressure. 

/"p° 
Again,  the  term  8nl    vd¥  must  be   positive    in  transferring 

S#  moles  from  the  saturated  solution  of  the  unstable  modifica- 
tion (osmotic  pressure  P0)  to  the  saturated  solution  of  the  stable 
modification  (osmotic  pressure  PjJ.  That  is,  P0  must  be  greater 


Transition  T 

Point 

FIG.  52. 

than  Pl5  or  the  saturation  concentration  (i.e.  solubility  which 
is  taken  as  proportional  to  osmotic  pressure  approximately)  of 
the  unstable  modification  is  greater  than  the  solubility  of  the 
stable  modification  at  the  given  temperature.  We  can  represent 
the  behaviour  of  the  vapour  or  saturated  osmotic  pressure  of  the 
two  modifications  a  and  f$  graphically  thus  (Fig.  52).  At  tempera- 
tures below  the  transition  point  the  a  modification  has  a  lower 
vapour  pressure  or  osmotic  pressure,  that  is,  the  a  is  the  stable 
modification  on  this  side  of  the  transition  point.  The  vapour 
pressure  and  osmotic  pressure  curves  are  naturally  continuous 
curves,  and  by  producing  them  beyond  the  transition  point  we 
see  that  at  temperatures  higher  than  the  transition  temperature 
the  j8  modification  has  now  the  lower  vapour  and  osmotic 
pressures,  that  is,  in  this  range  the  f$  is  the  stable  modification. 


ALLOTROPIC   CHANGE 


125 


(The  vapour  pressure  and  temperature  curves  will  in  general 
not  be  straight  lines  as  shown,  but  this  is  immaterial.) 


4.   The  Stable  Modification  has  the  Higher  Melting  Point. 

Let  us  denote  the  stable  modification  by  a  (Fig.  53), 
the  unstable  modification  by  j3,  the  fused  liquid  produced 
on  melting  by  y.  The  line  a,  a  denotes  the  vapour  pressure 
over  the  a  modification  ;  the  line  j8,  j3,  the  vapour  pressure 


I 


i 


T, 


Temperature 
FIG.  53. 

over  the  jS  modification ;  and  the  line  y,  y,  the  vapour  pres- 
sure over  the  liquid.  The  transition  point  of  a  into  )3 
lies  far  to  the  left  of  the  diagram ;  the  lines  a,  a  and  j8,  ft 
approach  one  another  at  lower  temperatures.  If  we  take 
any  temperature  Tj  it  will  be  seen  that  the  vapour  pressure 
over  the  unstable  solid  (j3)  is  greater  than  that  over  the  stable 
solid  (a).  The  position  N  denotes  the  melting  point  (melt- 
ing temperature)  of  the  stable  modification ;  M  the  melting 
point  of  the  unstable.  Hence  the  stable  phase  must  show  the 
higher  melting  point.  This  conclusion  rests  essentially  on  the 
statement  which  we  proved  in  Case  2,  namely,  that  the  vapour 
pressure  of  the  stable  modification  at  any  given  temperature 


126        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

is  lower  than  that  of  the  unstable  form.  ^The  diagram  also 
illustrates  that  ^stable  forms,  whether  solid  or  liquid^  have  a 
higher  vapour  pressure  than  the  stable  form  at  the  same 
temperature.  Thus  let  us  take  the  dotted  line  My.  This  is 
the  vapour  pressure  of  the  super-cooled  liquid,  i.e.  the  liquid 
form  should  have  changed  entirely  into  the  solid  modification 
j3  at  the  temperature  M,  but  it  has  not  done  so.  Hence  My 
represents  an  unstable  liquid  phase,  i.e.  unstable  with  respect  to 
the  j3  modification  and  hence  for  a  temperature  such  as  Tx  the 
vapour  pressure  of  the  unstable  super-cooled  liquid  is  higher 
than  that  over  the  solid  j3  modification  into  which  the  liquid 
should  have  been  transformed  at  M.  It  happens  that  the  solid 
P  modification  itself  is  unstable  with  respect  to  the  a  modifica- 
tion at  the  same  temperature  T1  so  that  at  T1}  if  there  is  any 
super-cooled  liquid  in  the  system  it  must  be  very  unstable 
indeed  with  respect  to  the  a  solid  modification.  On  the  other 
hand,  at  temperatures  above  M  the  liquid  form  is  stable  com- 
pared to  the  solid  /3  form,  as  is  shown  by  the  curve  M/2  (the 
vapour  pressure  curve  of  the  superheated  solid  /3  above  its 
melting  point)  lying  above  the  MN  curve,  which  gives  the 
vapour  pressure  of  the  liquid.  The  position  of  the  curve  NM 
shows,  however,  that  the  liquid  in  this  temperature  region  is 
unstable  with  respect  to  the  solid  modification  a,  as  can  be 
seen  by  considering  a  temperature  such  as  T2,  when  the  MN 
curve  of  the  liquid  lies  above  the  curve  for  the  solid  a  modifica- 
tion. At  the  melting  point  N  the  a  solid  and  the  liquid  are  in 
equilibrium.  At  temperatures  higher  than  this  the  liquid  is  the 
stable  form,  the  superheated  solid  a  being  unstable  as  is  shown 
by  the  relative  positions  of  the  curves  No,  (dotted)  and  Ny. 
Note  that  in  practice  solids  superheated  above  their  melting 
points  have  never  been  realised. 

As  a  matter  of  fact,  in  the  system  considered  we  are  dealing 
with  a  special  state  of  things  reached  when  transformation  is 
brought  about  rapidly.  What  happens  in  the  case  of  enantio- 
tropic  substances,  i.e.  those  which  show  a  transition  tempera- 
ture below  the  melting  point,  is  that  on  slowly  heating  up  the 
change  takes  place,  the  system  then  consisting  of  a  single  stable 
modification  (a)  which  possesses,  of  course,  a  single  melting 


ALLOTROPIC   CHANGE  127 

point,  i.e.  N.  By  carrying  out  the  heating  process  very  rapidly, 
however,  we  can  realise  such  a  point  as  M,  i.e.  we  can  cause 
the  unstable  variety  (j3)  to  melt  before  it  has  had  time  to 
change  over  into  the  stable.  These  equilibrium  relationships 
between  phases  will  not  be  discussed  further  until  we  take  up 
the  Phase  Law.  It  is  sufficient  for  the  present  to  have  proved 
the  statement  that  the  stable  modification  has  the  higher 
melting  point. 

A  further  case  of  equilibrium  between  different  crystalline 
forms  of  considerable  interest  may  be  mentioned,  namely,  that 
of  grey  tin  and  white  tin,  which  are  in  equilibrium  at  a  certain 
temperature  and  pressure  (the  transition  point).  The  reaction 
goes  practically  without  a  volume  change  at  a  given  tempera- 
ture and  equilibrium  is  reached  when  (8A)iv  =  °j  which  can  be 

ri 
put  in  the  form  /    vdp  =  o.      We  must  therefore  infer  that 

J  ii 

the  two  forms  are  in  equilibrium  when  their  vapour  pressures 
become  identical.     Further,  this  system  is  of  interest  because 
it  can  likewise  be  set  up  so  as  to  yield  electrical  energy.    Thus 
the  cell- 
White  tin  |  Tin  salt  solution  |  Grey  tin 

will  furnish  an  e.m.f.  At  the  transition  point  this  e.m.f.  is  zero, 
i.e.  white  and  grey  tin  are  in  equilibrium.  By  means  of  a 
cycle  we  can  show  that  this  latter  relation,  viz.  E  =  o,  must 
hold  at  the  point  at  which  the  vapour  pressures  over  the  white 
and  grey  tin  are  identical,  i.e.  the  transition  point.  Consider 
some  temperature  other  than  the  transition  point,  and  further 
consider  one  gram  equivalent  of  white  tin  to  be  vaporised,  the 
pressure  changed  from  pQ  to  /,  where  /'  is  the  vapour  pressure 
over  grey  tin  at  the  same  temperature,  and  then  the  vapour  to  be 
condensed  into  the  grey  tin.  The  work  done  up  to  this  point 

[PO 

is  /    vdp.     Assuming  the  white  tin  to  be  the  unstable  form, 
J  p' 

this  work  term  is  positive  work  done  by  the  system.  Now  let 
the  gram  equivalent  of  grey  tin  be  transferred  electrically  to 
the  white  tin  by  dissolving  off  the  grey  tin  electrode  and 
depositing  on  the  white,  i.e.  current  passes  through  the  cell 


p' 

or 


128       A    SYSTEM   OF  PHYSICAL   CHEMISTRY 

from  grey  tin  to  white.  If  the  e.m.f.  is  E  the  work  is  likewise  E. 
This  work  must  have  been  done  by  some  external  agency  ttpon 
the  system  itself,  for*the  natural  direction  in  which  the  system 
(the  cell)  tends  to  do  electrical  work  is  in  the  opposite  sense, 
i.e.  the  cell  itself  tends  to  work  in  such  a  way  as  to  dissolve 
the  unstable  white  form  and  precipitate  grey  tin,  so  that  eventu- 
ally there  will  be  nothing  but  grey  tin  present.  Since  E 
represents  work  done  by  the  surroundings,  then  —  E  represents 
the  work  done  by  the  system.  The  cycle  is  now  complete, 
and  since  it  is  isothermal  and  reversible,  the  total  work  must 
be  zero  by  the  Second  Law.  That  is — 

'Po 

f  (-E)=o 

PO 

E=  /   vdp 

J  P' 

In  general  we  can  say  that  the  electrical  energy  is  just 
equal  to  the  three-stage  distillation  work.  Now  at  the  tran- 
sition point — 

IP* 

/0  =/       .*.  /   vdp  =  o      .-.  E  =  o 
J  pf 

We  reach  the  same  conclusion  of  course  by  following  out  the 
very  similar  type  of  reasoning  involved  in  the  principle  of 
virtual  work. 

Suppose  the  grey  tin  and  white  tin  are  in  equilibrium. 
Then  in  the  transfer  of  8n  moles  from  grey  to  white  the  work 
at  constant  temperature  and  volume  is  zero.  Carrying  the 
process  out  by  the  distillation  method  we  see  that  at  equi- 

/grey. 
vdp  —  o,    or    the   vapour    pressures    over   the 
white 

modifications  are  identical.  Also  at  the  equilibrium  point 
suppose  a  quantity  8n  moles  transferred  electrically  through 
the  solution  from  the  grey  to  the  white  tin  electrode.  Suppose 
the  e.m.f.  is  denoted  by  E  and  the  electrochemical  equivalent l 

1  The  electrochemical  equivalent  of  a  substance  is  here  defined  as  the 
mass  in  moles  of  the  substance  with  which  one  faraday  of  electricity  is 

associated  when  the  substance  is  in  the  ionic  form.     The  expression  — 

Ofa 

denotes  therefore  the  quantity  of  electricity  associated  with  8«  moles. 


THE   FUNCTION  <l>  129 

of  tin  by   tfsn,  then   the   electrical  work  in  volt-faradays   is 

C\ 

-  X  E.     This  must  be  zero.     Since  8u  is  a  positive  term 

#Sn 

and  #sn  represents  a  positive  quantity,  it  follows  that  E  ==  o, 
which  was  to  be  shown. 


THE  FUNCTION  0. 

One  has  to  recognise  that  the  above  attempt  to  get  at 
general  thermodynamic  criteria  for  systems  in  equilibrium  by 
simply  applying  the  principle  of  work  is  somewhat  limited  in 
application.  To  treat  the  subject  of  general  criteria  it  is 
necessary  to  pursue  a  more  analytical  method  of  presenting 
thermodynamic  relations  such  as  that  followed  to  some  extent 
in  Chapter  II.  For  those  who  have  read  this  chapter  we  can 
take  up  the  question  of  equilibrium  in  the  following  way. 

We  saw  that  for  a  reversible  change  in  any  system,  we 
could  write  in  general  the  increase  in  the  total  energy  d\J  in 
the  following  way — 

dU  =  Td<j>  —  pdv  —  dw 

Where  Td(/>  represents  the  increase  in  the  heat  content  (the 
heat  absorbed),  pdv  represents  the  mechanical  external  work 
done,  and  du>  any  other  form  of  external  work  which  may 
also  have  been  done,  e.g.  output  of  electrical  energy,  capillary 
energy,  radiant  energy,  gravitational  work,  etc.  For  practical 
purposes  the  term  dw  may  be  used  to  denote  output  of 
electrical  energy.  Since  the  process  we  are  considering  is 
reversible,  all  the  work  terms  are  maximum  work  terms. 
Now,  from  the  above  equation,  since  Td(j)  must  be  equal  to 
d(T(f>)  —  $dtt  it  follows  that— 

dU  —  d(T<l>)  =  —  </>dt  —pdv  —  dw 
or  d(U  —  T</>)  =  —  <t>dt  —pdv  —  dw. 

The  quantity  U  represents  the  total  energy  of  the  system, 
the  quantity  T<£  is  called  by  Helmholtz  the  bound  or  unavail- 
able energy,  so  that  the  difference  of  the  two,  namely  (U  —  T</»), 
may  be  called  the  free  energy  (Helmholtz)  and  denoted  by/, 
T.P.C. — n.  K 


130       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

This  is  the  strict  definition  of/.  We  can  thus  write  the  above 
equation  — 

nf=  —  $dt  —pdv  —  dw 

Now  if  the  process  (reaction)  which  we  are  considering  is 
one  that  goes,  i.e.  can  be  made  to  go,  reversibly  at  constant 
temperature  and  volume,  the  terms  $dt  and  pdv  become  zero, 
and  the  change  of  /is  represented  by— 


(Note  the  negative  sign  denotes  that  positive  work  output 
by  the  system  means  decrease  in  free  energy.)  If  we  take 
as  our  standard  of  equilibrium,  that  equilibrium  is  reached 
when  dw  =  o  ;  then  (<2^)TV  =  o.  This  is  the  same  result  we 
arrived  at  before,  namely,  in  a  reaction  such  as  that  in  the 
Daniell  cell,  which  goes  reversibly  at  constant  temperature 
and  practically  constant  volume,  and  reaches  an  equilibrium 
when  the  e.m.f.  is  zero.  Consider  such  a  system  to  have 
reached  the  equilibrium  point,  then  we  can  say  that  if  we 
imagine  a  change  in  either  direction  such  change  would  be 
accompanied  by  an  increase  in  free  energy  at  constant  tem- 
perature and  volume,  and  would  not  therefore  occur.  The 
free  energy  of  the  system  as  a  whole  is  a  minimum,  and 
the  free  energy  of  the  reactants  is  equal  to  the  free  energy 
of  the  resultants  at  the  equilibrium  point. 
Now  returning  to  the  equation— 

—  T<     =  —    dt  —   dv  —  dw 


we  can  further  transform  this  by  writing  — 
pdv  =  d  (pv)  —  vdp 
into  the  relation  — 

—  dw 


This  expression  (U  —  T</>  -\-  pv)  is  denoted  by  0.  If  we 
consider  a  reaction  made  to  go  reversibly  at  constant  tempera- 
ture and  pressure  (such  as  vaporisation  of  water),  we  can  see 
that  any  possible  change  in  0  can  be  expressed— 


PHASE   EQUILIBRIA  131 

We  take  as  our  standard  of  equilibrium  in  a  reaction  which 
goes  with  a  volume  change,  but  can  be  made  to  go  at  constant 
temperature  and  pressure,  that  equilibrium  is  reached  when 
(<#£)xp  =  o  ;  that  is  again  when  dw  is  zero.  In  the  case  of 
water  vaporising  the  term  dw  (electrical  work)  does  not  appear 
at  all,  and  the  equilibrium  is  reached  at  a  given  temperature, 
and  a  certain  pressure,  namely,  the  pressure  of  saturated 
vapour.  Water  and  water  vapour  are  in  equilibrium  when 
the  0  of  i  gram  of  water  is  equal  to  the  0  of  i  gram  of 
vapour,  and  this  occurs  when  the  vapour  is  saturated.  At  the 
freezing  point  the  0  of  i  gram  of  ice  =  the  0  of  i  gram 
of  liquid  water,  because  the  change  can  go  at  constant  tempera- 
ture and  pressure.  It  can  likewise  go  at  constant  temperature 
and  volume,  and  hence  (df)  =  o  also.  At  the  transition 
point  the  0  of  i  gram  of  rhombic  sulphur  =  the  0  of  i  gram 
of  monoclinic  sulphur,  and  similarly  the  0  of  i  gram  of 
grey  tin  =  the  0  of  i  gram  of  white  tin.  The  importance 
of  0  as  defining  equilibrium  between  phases  (solid  and  solid 
(rhombic  and  monoclinic  sulphur),  solid  and  liquid  (ice  and 
water),  liquid  and  vapour  (water  and  water  vapour))  will  be 
apparent  when  we  come  to  deduce  the  Phase  Rule  of  Gibbs. 


CHAPTER   V 

Chemical  equilibrium  in  homogeneous  systems,  from  the  thermodynamic 
standpoint — Gaseous  systems — Deduction  of  the  law  of  mass  action 
— The  van  't  Hoff  isotherm — -Principle  of  "  mobile  eqxiilibrium  " 
(Le  Chatelier  and  Braun) — Variation  of  the  equilibrium  constant 
with  temperature. 

VAN  'T  HOFF'S  "  EQUILIBRIUM   Box  " 

AN  "  equilibrium  box  "  is  a  vessel  of  unchangeable  volume  in 
which  the  various  substances  taking  part  in  a  given  reaction 
are  present  in  the  equilibrium  state.  The  walls  of  this  vessel 
are  suitably  permeable  to  given  substances  and  impermeable 
to  others.  It  is  only  possible  to  apply  the  equilibrium  box 
idea  to  a  reaction  taking  place  in  a  homogeneous  system,  e.g. 
gases  and  dilute  solutions.  Let  us  at  present  confine  our  con- 
sideration to  a  given  reaction,  viz,  A  -f  B  =  C  +  D.  In  the 
box  the  four  substances  A,  B,  C,  D  are  present  in  equilibrium. 
Let  us  introduce  isothermally  and  reversibly  i  mole  of  A 
and  i  mole  of  B  into  the  box  (through  sides  permeable  only 
to  each  of  these),  at  the  equilibrium  concentration  or  pressure, 
and  at  the  same  time  remove  i  mole  of  C  and  i  mole  of 
D  (through  sides  permeable  to  these),  also  at  the  equilibrium 
concentration  or  pressure.  What  has  happened  is  the  trans- 
formation of  A  and  B  into  C  and  D  at  the  equilibrium  con- 
centration or  pressure  conditions,  without  the  expenditure  of 
mechanical  work  since  the  volume  of  the  box  is  constant. 
Further,  the  system,  i.e.  the  box  and  its  contents,  is  at  the  end 
in  exactly  the  same  state  as  at  the  beginning,  and  this  is  true 
for  all  stages.  No  actual  external  work  has  been  done  in  the 
chemical  transformation  under  these  conditions.  If  A  and  B 
are  initially  at  an  arbitrary  pressure  or  concentration,  work  will 
be  done  either  by  or  upon  them  in  bringing  them  isothermally 


THE  LAW   OF  MASS   ACTION 


133 


and  reversibly  to  the  concentration  corresponding  to  equili- 
brium. When  this  stage  has  been  reached  they  may  be  intro- 
duced into  the  equilibrium  box  where  they  might  be  imagined 
to  react  giving  rise  to  C  and  D,  which  can  be  removed  at  the 
equilibrium  concentration  value,  no  work  being  done  in  doing 
this  "  in  and  out  "  operation. 


THERMODYNAMIC  DEDUCTION  OF  THE  MASS  ACTION  EX- 
PRESSION FOR  EQUILIBRIUM  IN  A  HOMOGENEOUS  GASEOUS 
SYSTEM. 

We  can  do  this  by  means  of  an  isothermal  reversible  cycle. 
The  proposition  we  make  use  of  is  that  the  sum  of  all  the  work 
terms  for  a  completed  isothermal  cycle  add  up  to  zero  by  the 
Second  Law.  Suppose  the  reaction  is— 

A  +  B   ^  C  +  D 

Suppose  we  have  two  reservoirs  (Fig.  54)  each  containing 
four  substances,  A,  B,  C,  D,  in  equilibrium  ^  the  absolute  con- 


I 


-/ 


r  ~ 


v 


n 


FIG.  54. 

centration  terms  in  the  one  being  CA,  CB,  Cc,  CD,  and  in  the 
other  yA,  ye,  yc,  yo,  each  reservoir  is  an  equilibrium  box.  The 
temperature  is  the  same  for  both  boxes. 

First  Step. — Take  out  i  mole  of  A  from  reservoir  I  isother- 
mally  and  reversibly.  In  drawing  out  i  mole  of  A  at  the  partial 
pressure  it  has  in  I,  say  ^AO,  the  system  does  work  /AO?'AO, 
where  ?'AO  is  the  volume  of  i  mole  of  A  at  partial  pressure /AO. 


134       A    SYSTEM   OF  PHYSICAL    CHEMISTRY 

First  Sub-step. — Now  alter  #AO  to  v±>  reversibly,  the  pres- 

/^A* 
pdv. 

The  mole  is  now  at  the  partial  pressure  possessed  by  the  same 
substance  A  in  vessel  II. 

Second  Sub-step  with  A. — Isothermally  compress  this  mole 
into  reservoir  II.,  work  done  upon  gas  =  +/A(^A',  and  there- 
fore work  done  by  the  gas  is  — /A'^A'.  The  first  step  plus  these 
sub-steps  we  have  already  seen  simply  add  up  to  the  expression 

v&&. 

PA' 

Second  Step. — Along  with  the  transfer  of  A  we  suppose  a 
transfer  of  i  mole  of  B  from  I.  to  II.  to  take  place.  Work 


/•^BO 
per  mole  of  B  =  /     \ 


Since  the  system  is  gaseous,  let  us  assume  the  gas  laws,  and 
we  can  write — 

/AO  =  RTCA  and /A'  =  RTyA 

and  hence — 

P 

work  term  for  A  =  RT  log  - 

/-i 

work  term  for  B  =  RT  log 

VB 

Third  Step. — Now  in  this  equilibrium  box,  viz.  reservoir  II., 
suppose  the  A  and  B  we  have  added  are  changed  into  C  and 
D,  no  external  work  being  done. 

Fourth  Step. — Now  transfer  i  mole  of  C  and  i  mole  of  D 
from  II.  to  I. 

Work  =  RT  log  £'  +  RT  log  ^ 

Cc  L'D 

and  let  these  molecules  of  C  and  D  change  into  A  and  B  in 
reservoir  I.,  no  work  being  done  in  either  equilibrium  box 
during  the  cycle  as  a  whole.  The  cycle  is  now  completed,  the 
initial  conditions  being  restored.  Since  this  has  been  done 


THE   LAW  OF  MASS  ACTION  135 

isothermally  and  reversibly  by  the  Second  Law  of  Thermo- 
dynamics, no  work  on  the  whole  has  been  done.  That  is — 

RT  log  -A+  RT  log  ^  +  RT  log  ~+  RT  log  ^  =  o 

7A  7fi  UD 

or 

log  CA  +  log  CB  —  log  Co  —  log  CD 

=  l°S  7A  +  log  yB  —  log  yc  —  log  yD 

C^==V^»=  constant 
Cc  X  CD      yc  X  yD 

the  term  "constant"  is  justified,  for  no  numerical  relation 
whatsoever  was  assumed  to  exist  between  C's  and  y's  beyond 
the  fact  that  they  corresponded  in  each  reservoir  to  equilibrium 
at  the  same  temperature.  Hence  we  obtain  the  Guldberg 
Waage  expression,  or  the  Law  of  Mass  Action  for  a  system 
obeying  the  gas  laws. 

Note  to  Preceding  Proof. 

In  working  with  an  equilibrium  box,  in  order  that  equilibrium  shall 
always  prevail,  we  must  not  alter  the  concentration  of  any  one  of  the 
components.  That  is,  as  we  introduce  the  reactants  (say  into  vessel  II.), 
we  must  simultaneously  remove  an  equivalent  quantity  of  resultants  (which 
might  be  regarded  as  actually  the  transformed  reactants  which  we  are 
putting  in).  Similarly,  in  the  case  of  vessel  I.  the  addition  and  subtraction 
must  be  simultaneous.  The  cycle,  though  described  as  a  series  of  consecu 
tive  effects,  must  really  be  imagined  as  taking  place  "  everywhere  at  once," 
that  is,  disregarding  the  time  factor  in  order  that  each  equilibrium  box  may 
be  at  every  moment  a  system  possessing  equilibrium  concentration. 

Another  point  requires  mention.  In  the  case  chosen  the  reaction 
involved  no  change  in  the  number  of  molecules,  i.e.  no  increase  in  volume 
at  constant  pressure  or  increase  in  pressure  at  constant  volume.  It  might 
seem,  therefore,  that  the  absence  of  work  in  the  transformation  from  re- 
actants to  resultants  (or  vice  versa]  in  the  equilibrium  boxes  really  depended 
on  this  property  of  the  system.  This  is  not  the  case,  however.  The  con- 
ception of  an  equilibrium  box  and  the  principle  of  virtual  work  are  quite 
as  applicable  to  reactions  which  involve  a  change  in  the  number  of  mole- 
cules. Thus  take  the  case  of  the  combination  of  2  volumes  of  hydrogen 
with  i  volume  of  oxygen  to  form  2  volumes  of  water  vapour.  In  the 
process  going  in  this  direction  there  is  a  decrease  of  I  volume,  but 
under  the  equilibrium  box  conditions  and  arrangement  this  loss  of  I  volume 
in  the  box  is  exactly  balanced  by  the  fact  that  we  put  in  3  volumes  and 


136       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

only  took  out  2.  The  contents  of  the  box  remain  as  they  were,  the  volume 
of  the  box  remaining  constant.  Of  course  in  all  cases  there  will  be  a 
positive  or  negative  heaf  passing  out  from  the  box  into  the  constant 
temperature  reservoir  with  which  it  is  in  contact  ;  such  heat  being  the 
equivalent  of  the  total  energy  liberated  or  absorbed  in  the  reaction,  but  no 
work  comes  out  or  goes  in.1 

The  Law  of  Mass  Action,  as  applied  to  homogeneous  gaseous  systems, 
has  already  been  illustrated  in  Part  I.  (Vol.  I.)  by  several  concrete  examples. 
It  is  unnecessary,  therefore,  to  repeat  or  extend  these  in  the  present  dis- 
cussion of  the  law,  as  sufficient  instances  of  its  experimental  verification 
have  already  been  given. 

As  already  pointed  out  in  dealing  with  the  Law  of  Mass  AcMon  from 
the  kinetic  standpoint,  the  principle  holds  good  in  the  first  instance  for 
homogeneous  gaseous  systems.  It  can  likewise  be  extended  to  the  case  of 
heterogeneous  equilibrium,  in  which  a  gas  or  gases  are  essential  to  the  re- 
action, e.g.  the  dissociation  of  calcium  carbonate,  or  ammonium  chloride. 
Every  solid  or  liquid  possesses  a  vapour  pressure,  however  small,  and 
hence  in  the  vapour  space  above  the  solid  calcium  carbonate  and  lime  we 
have  some  gaseous  CaCO3  molecules,  gaseous  CaO  molecules,  and  a  large 
quantity  of  gaseous  CO2  molecules.  This  space  containing  vapour  can  be 
treated  as  a  homogeneous  system  in  which  a  gaseous  reaction  can  take 
place,  and  the  equilibrium  is  determined  by  — 

Ceo  2  X  Ccao 


gas 

The  simplification  which  can  be  introduced  here,  however,  is  that  the 
concentration  terms  for  gaseous  carbonate  and  lime  are  themselves  constant 
(at  constant  temperature),  since  the  solids  are  present  and  the  correspond- 
ing vapours  are  therefore  saturated.  The  equilibrium  in  such  a  case  is 
thus  defined  by  the  CO2  pressure  alone,  since  this  is  not  "saturated,"  i.e. 
no  solid  or  liquid  CO2  is  present.  That  is  — 

K  =  K'Cco2 

or  Cco2  —  constant  at  constant  temperature. 

1  From  the  purely  thermodynamical  standpoint  it  is  interesting  to  note 
that  the  change  from  reactants  to  resultants,  even  as  it  occurs  in  the 
equilibrium  box,  must  be  an  irreversible  process  because  it  occurs  naturally 
or  spontaneously  ;  but,  regarded  as  a  link  in  the  chain  of  operations,  the 
reverse  change  occurs  in  the  box  where  the  taking  in  and  putting  out 
operations  are  exchanged.  Chemical  reversibility  is  not  the  same  thing  as 
thermodynamical  reversibility.  There  is  really  a  difficulty  here,  however. 


THE    VAN  >T  HOFF  ISOTHERM  137 

THE  VAN  'T  HOFF  ISOTHERM. 

This  is  an  expression  which  gives  the  maximum  external 
work  which  can  be  obtained  from  a  given  chemical  process  occur- 
ring in  a  gaseous  system  at  constant  temperature^  the  work  being 
expressed  in  terms  of  the  equilibrium  constant.  Later  we  shall 
see  the  significance  of  this  "  Isotherm  "  as  a  measure  of  the 
affinity  of  a  reaction.  For  the  present  the  expression  will  be 
deduced  by  the  aid  of  thermodynamics  (and  the  equilibrium 
box),  as  we  require  the  isotherm  in  the  subsequent  con- 
sideration of  the  variation  of  the  equilibrium  constant  K  with 
temperature. 

Deduction  of  the   Van  V  Hoff  Isotherm. 

Let  us  suppose  that  we  are  dealing  with  the  reaction — 

2H2-j-O2  =  2H2O 

taking  place  in  the  gaseous  state.  The  arbitrarily  chosen 
concentration  value  of  the  hydrogen  and  oxygen  are  C7H2  and 
C'o2,  and  we  want  to  find  what  will  be  the  maximum  work  out- 
put involved  in  making  2  moles  of  hydrogen  and  i  mole  of 
oxygen  react  (thermodynamically,  i.e.  reversibly)  so  as  to  end  up 
finally  with  a  system  consisting  of  water  vapour  at  an  arbitrary 
concentration  C'H2o-  We  imagine  that  there  are  three  practi- 
cally infinitely  large  vessels  (Fig.  55)  containing  respectively 
H2,  O2,  and  H2O  gases  at  the  arbitrary  concentration, 
C'H2,  C'o2,  C'HSO.  The  temperature  is  T  throughout.  The  size 
of  these  three  vessels  is  so  great  that  on  withdrawal  or  addition 
of  i  mole  or  even  2  moles  of  the  constituent,  this  can  be  done 
without  altering  the  concentration  in  the  vessel,  i.e.  the  mole 
of  gas  can  be  drawn  off  at  constant  pressure.  Besides  these 
three  large  vessels,  we  likewise  suppose  that  we  have  at  our 
disposal  an  equilibrium  box  containing  H2,  Go,  and  H2O 
gases  at  the  equilibrium  concentrations  CeH2,  Ceo2,  CeH2o5 
corresponding  also  to  the  temperature  T.  One  face  of  the 
equilibrium  box  possesses  the  property  of  being  permeable  to 
hydrogen  and  impermeable  to  oxygen  or  water.  A  second 
face  is  only  permeable  to  oxygen,  a  third  face  permeable  to 


138        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

water  vapour  only.     Now  we  suppose  the  following  thermo- 
dynamic  process  earned  out. 

I.  By  means  of  a  cylinder  and  piston  one  can  take  2  moles 
of  hydrogen  from  the  stock  at  concentration  CH  to  the 
equilibrium  box  (where  hydrogen  is  at  concentration  C€H2) 
reversibly  and  isothermally,  thereby  accomplishing  a  three- 
stage  process  in  which 

..initial  pressure  in  infinitely  large  vcssol 

Aper  2  moles  of  rr£  =  2  I  vdp  (where  v  ==  molecular  volume) 

*  final  pressure  in  equilibrium  box 


Reservoir  of 
practically 
infinite  size 
containing 
Hydrogen 


Reservoir  of 
practically 
infinite  sizs 
containing 
H20  vapour 


w 


The  direction  of  the 
arrows  illustrates 
the  thermodynamlc 
method  of  carrying 
out  the  reaction. 

2Ho+Oo-*2HoG 


Equilibrium 
Box 
C 


-i 


Reservoir  of 
practically 
infinite  size 
containing 
Oxygen 

atc'o2 


FIG.  55. 


The  gas  is  supposed  to  obey  the  gas  law,  so  that  we  can 

evidently  write 

f-\i 

Aper  2  moles  of  H2  =  2RT  log       ~ 


the  number  "  2  "  entering  in  because  2  gram-molecules  have 
been  transferred.  C'H,  is  the  initial  concentration,  C£H2  the 
final  concentration  reached  in  this  process. 


THE    VAN  >T  HOFF  ISOTHERM  139 

II.  Simultaneously  with  operation  I.  we  suppose  i  gram 
mole  of  oxygen  is  taken  from  the  stock  vessel  at  C'o2  to  the 
equilibrium  box  at  Ceo2-     The  work  done  is  given  by — 

C'o 

Aper  1  mole  of  02  =  RT  log  ^  - 

The  hydrogen  and  oxygen  now  in  the  equilibrium  box 
react  without  doing  external  work,  water  vapour  being  removed 
at  the  same  time. 

III.  Two  moles  of  water  vapour  are  removed  from  the 
equilibrium  box  at  concentration  CeH2o,  and  brought  to  the 
vessel   at    arbitrary   concentration    C'HOO.      The   initial   con- 
centration in  this  case  is  Cen2o,  the  final  concentration  being 
C'H2o.      Hence  the  work  is  given  by — 

Aper  2  moles  of  HaO  gas  =  2RT  log  •—?-* 

The  total  process  which  has  been  carried  out  isothermally 
and  reversibly  is  simply  the  chemical  combination  of  2  moles 
of  H2  and  i  mole  of  O2  at  a  given  arbitrary  concentration  to 
produce  H2O  vapour  also  at  arbitrary  concentration.  If  the 
total  maximum  work  or  decrease  in  free  energy  is  A,  then — 

A  =  Aper  2  moles  of  ~T~  Aper  mole  of  ~~j~~  Aper  mole  of  water 
hydrogen  oxygen  vapour 

that  is — 

A  =  2RT  log  ^'H2  +  RT  log  ^+  2RT  log  ^Ea2j 

CeH2  Ce02  C  H20 

But  the  equilibrium  constant  K  of  the  H2O,  O2,  H2  reaction 
(2H2  -j-  O2  ->  2H2O),  at  temperature  T,  is — 

C2 

K===c60  x^CeH 

or  log  K  =  log  C2H20  -  log  Ceo2  -  log  C2eHa 

Hence  the  above  expression  for  A  may  be  written — 


It  will   be  observed  that   the   magnitude   of  A   depends 


HO        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

on  how  far  the  arbitrary  chosen  concentration  of  the  system 
is  from  the  equilibrium  concentration  state.  If  the  arbitrary 
chosen  state  happens  to  correspond  to  the  equilibrium  state 
naturally  no  work  will  be  done  in  reaching  the  equilibrium 
state.  This  the  above  formula  expresses.  It  will  also  be 
observed  that  the  above  formula  is  deduced  on  the  assumption 
that  the  system  obeys  the  gas  laws.  Numerical  illustration  of 
this  formula  will  be  given  when  we  come  to  study  the  affinity 
of  a  reaction.  The  most  general  mode  of  expressing  the 
isotherm  when  several  substances  react,  e.g. — 

"IAI  +  ^A2  +  etc.  =  i/A;  +  i/ A;  +  etc. 

is  simply  to  write  the  arbitrary  concentration  term  in  an 
abbreviated  form,  viz.  RTZV  log  C,  where  C  stands  for  all  the 
arbitrary  concentration  terms  (the  reactants  in  the  denomi- 
nator, the  resultants  in  the  numerator  in  the  log.  term),  and  27v 
simply  stands  for  v\  +  v'z  +  etc-  —  VL  ~  vz  ~  etc->  so  tnat  tne 
general  form  of  the  isotherm  is — 

A  =  RT  log  K  —  RT£i>  log  C 


THE  PRINCIPLE  OF  LE  CHATELIER  AND  BRAUN,  OR  "THE 
PRINCIPLE  OF  MOBILE  EQUILIBRIUM." 

This  generalisation  was  first  given  by  Le  Chatelier  in 
1884,  and  later  independently  by  Braun  in  1887-88.  No 
"proof"  in  the  ordinary  sense  can  be  given  of  it;  it  is  a 
generalisation  based  on  experience.  It  has  been  applied  in 
all  branches  of  Physics  and  Chemistry,  and  has  shown 
itself  to  be  a  valid  and  perfectly  general  law  of  nature. 
Since  it  involves  no  assumptions  regarding  molecular  structure 
of  systems,  it  is  to  be  regarded  as  a  thermodynamical  law.  As 
a  matter  of  fact,  it  is  comprehended  in  the  Second  Law  of 
Thermodynamics.  This  connection  is  discussed  very  fully  in 
Chwolson's  Textbook  of  Physics  (vol.  iii.  p.  474,  seq^  to  which 
the  reader  is  referred  for  further  information  on  this  point. 

The  principle  may  be  stated  as  follows  : — 

WHEN  A  FACTOR  DETERMINING  THE  EQUILIBRIUM  OF  THE 


PRINCIPLE   OF  MOBILE   EQUILIBRIUM        141 

SYSTEM  IS  ALTERED,  THE  SYSTEM  TENDS  TO  CHANGE  IN  SUCH  A 
WAY  AS  TO  OPPOSE  AND  PARTIALLY  ANNUL  THE  ALTERATION 
IN  THE  FACTOR.  THE  SAME  IDEA  IS  CONVEYED  BY  SAYING 
THAT  EVERY  SYSTEM  IN  EQUILIBRIUM  IS  CONSERVATIVE,  OR 
TENDS  TO  REMAIN  UNCHANGED.  THAT  IS,  CONSIDERING  A 
PHYSICAL  OR  CHEMICAL  SYSTEM  IN  EQUILIBRIUM,  THE  EQUI- 
LIBRIUM BEING  FIXED  BY  THE  NATURE  OF  THE  SYSTEM  AND 
CONDITIONS  SUCH  AS  TEMPERATURE  AND  PRESSURE,  THE  PRIN- 
CIPLE STATES  THAT  IF  WE  ALTER  ONE  OF  THESE  CONDITIONS 
OR  PARAMETERS,  SAY  THE  TEMPERATURE,  THE  SYSTEM  WILL 
CHANGE  IN  SUCH  A  DIRECTION  AS  TO  TEND  TO  ANNUL  THIS 

CHANGE  IN  TEMPERATURE.  The  principle  will  be  made  clearer 
by  a  few  examples. 

Consider  the  case  of  a  gas  occupying  a  certain  volume  at 
a  given  pressure  and  temperature.  Suppose  the  pressure  is 
increased.  The  volume  is  thereby  decreased,  and  according 
to  the  principle  the  system  ought  to  oppose  this  volume 
decrease.  It  does  so  in  rising  in  temperature,  for  a  rise  in 
temperature  tends  to  make  the  volume  increase.  We  are 
familiar  with  the  fact  that  by  compressing  a  gas  its  tem- 
perature rises.  This  behaviour  is  in  agreement  with  the 
principle.  Take  as  another  case  a  gaseous  system  such  as 
iodine  vapour  partially  decomposed  and  in  equilibrium  with 
its  dissociation  products  at  a  given  temperature  and  pressure. 
Now  suppose  the  temperature  of  the  system  is  raised.  Accord- 
ing to  the  principle  the  system  will  change  so  as  to  tend  to 
annul  this  rise  in  temperature.  What  actually  happens  is  that 
the  dissociation  of  iodine  molecules  increases,  and  separate 
experiments  show  that  a  gaseous  dissociation  is  accompanied 
by  an  absorption  of  heat.  In  a  process  involving  heat 
absorption  the  temperature  would  naturally  fall,  and  this 
fall  of  course  opposes  the  rise  in  temperature  impressed  upon 
the  system  by  the  external  agency  in  the  first  instance.  The 
fact  that  gaseous  dissociation  increases  \\it\\rise  in  temperature 
is  thus  directly  due  to  the  fact  that  the  process  of  dissociation 
involves  heat  absorption.  In  cases  of  dissociation  involving 
an  evolution  of  heat,  the  principle  leads  us  to  expect  that 
an  increase  of  temperature  will  cause  a  decrease  in  the 


142        A   SYSTEM  OF  PHYSICAL    CHEMISTRY 

dissociation,  i.e.  a  recombination  of  the  dissociation  products, 
since  in  this  processj^eat  is  absorbed. 

What  has  been  said  in  connection  with  dissociation  holds 
for  all  kinds  of  chemical  reactions  either  homogeneous  or 
heterogeneous.  An  endothermic  reaction  will  be  favoured 
by  a  rise  in  temperature,  an  exothermic  reaction  by  a  lowering 
of  temperature.  The  effect  produced,  say,  by  increasing  or 
decreasing  the  pressure  on  a  gaseous  equilibrium  can  also  be 
predicted  by  the  principle,  provided  we  know  whether  the 
reaction  considered  involves  a  volume  increase  or  decrease. 
In  the  case  of  the  hydriodic  acid  dissociation  which  involves 
no  change  in  volume  the  effect  of  pressure  ought  to  be  nil 
according  to  the  principle,  and  this  is  in  agreement  with  experi- 
ment as  we  have  already  seen  in  Part  I.  (Vol.  I).  In  the  case 
of  the  dissociation  process  of  I2->2l,  the  volume  increases. 
Now  if  we  increase  the  pressure  the  system  will  change  in  the 
chemical  sense  so  as  to  annul  this  increase  in  pressure.  It 
can  do  so  by  the  dissociation  diminishing,  as  thereby  there  are 
fewer  molecules  present,  and  the  pressure  of  the  system  itself 
tends  to  be  less  than  in  the  first  instance.  Notice  that  we  are 
only  employing  the  term  "  tends  to  be  less."  Actually  the 
pressure  of  the  system  in  the  final  state  is  greater  than  in  the 
initial  state,  but  it  is  not  so  great  as  it  would  have  been  if 
the  extent  of  the  dissociation  had  remained  unchanged.  The 
behaviour  in  this  case  is  to  be  distinguished  from  that  of  a 
single  gas  in  which  no  chemical  reaction  was  possible,  and  in 
which  we  consider  changes  in  pressure,  temperature,  and 
volume  simultaneously.  Of  course  in  the  dissociation  case 
also  the  temperature  will  tend  to  rise  on  compression  if  this 
be  rapid,  but  after  allowing  sufficient  time  to  elapse  so  that 
the  temperature  is  the  same  as  at  the  start,  the  dissociation  will 
be  found  to  have  permanently  diminished  as  already  described. 
Instead  of  confining  our  illustrations  to  gases,  let  us  consider 
a  heterogeneous  reaction  involving  solid  and  liquid  phases, 
e.g.  the  process  of  solution  of  a  solid  in  a  solvent.  Suppose 
we  have  a  saturated  solution  of  a  substance,  and  some  of  the 
undissolved  solid  in  contact  with  it  at  a  given  temperature. 
Now  suppose  the  temperature  is  raised.  Will  the  substance 


THE    VAN  >T  HOFF  ISOCHORE  143 

still  further  dissolve  or  will  there  be  a  precipitation  of  the 
substance  already  in  solution?  This  depends  entirely  on 
whether  the  process  of  solution  of  the  solid  involves  a  heat 
evolution  or  absorption.  If  the  solid  dissolves  with  absorption 
of  heat,  the  raising  of  the  temperature  will  cause  some  more 
of  the  solid  to  dissolve  since  this  tends  to  annul  the  tempera- 
ture rise.  For  such  a  substance  the  solubility  increases  with 
the  temperature.  If  the  solid  dissolves  with  evolution  of  heat, 
a  rise  in  temperature  will  cause  some  of  the  solid  to  be  pre- 
cipitated from  the  solution,  which  at  this  higher  temperature 
is  now  super-saturated,  for  by  this  process  (the  reverse  of 
solution)  heat  is  absorbed,  and  therefore  the  temperature  rise 
is  partly  annulled,  at  least  temporarily.  Again  consider  the 
same  system  and  suppose  the  pressure  upon  it  increased 
at  constant  temperature.  The  system  will  change  in  the 
following  way.  Further  solution  of  the  solid  will  take  place 
if  the  process  of  solution  is  accompanied  by  a  contraction  in 
volume  of  the  system  as  a  whole,  as  the  contractions  tends  to 
annul  the  increase  in  pressure  applied  by  a  gas,  say  the  atmo- 
sphere. On  the  other  hand,  some  of  the  dissolved  solid  will  be 
precipitated  from  solution  if  this  process  involves  a  volume 
decrease  (a  contraction). 

The  Variation  of  the  Equilibrium  Constant  with  Temperature. 
The  van  V  Hoff  Isochore. 

The  expression  we  are  about  to  deduce  is — 

d  log_K  _        JJ_ 
dT  RT2 

vhere  K  is  the  Mass  Law  equilibrium  constant,  U  the 
ecrease  in  total  energy  involved  in  the  transfer  of  the 
ecessary  number  of  moles  of  reactants  (required  by  the  stoich- 
ometric  or  ordinary  chemical  equation)  into  resultants,  T  the 
bsolute  temperature,  R  the  gas  constant  per  mole.  This 
xpression  is  known  as  the  van  V  Hoff  Isochore,  and  is  simply 
he  Le  Chatelier-Braun  principle  applied  quantitatively  (by 
naking  use  of  the  First  and  Second  Laws  of  Thermodynamics) 
o  the  case  of  the  effect  of  temperature  on  the  equilibrium 


144        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

constant    K   of  a   given   reaction.     Suppose,  for  simplicity's 
sake,  we  consider  a  gaseous  system  in  which  the  reaction  is  — 

i^Ai  +  v2A2  -j-  etc.  =  i/A^  -f  i/A^  +  etc. 
The  equilibrium  constant  K  is  — 


1/VI"'  X  [A21"2  X  .  .   . 

Suppose  we  have  a  number  of  vessels  each  containing  the 
solutions  A!,  A2,  A'j,  A'2,  at  arbitrarily  chosen  concentration 
values.  Then  the  work  done  in  bringing  the  correct  stoich- 
iometric  quantities  (as  required  by  the  chemical  equation),  i.e. 
v±  moles  of  A1}  v2  moles  of  A2,  etc.,  into  the  "  equilibrium 
box  "  and  transforming  them  into  resultants,  and  these  being 
brought  to  the  arbitrarily  chosen  concentration  values,  is  given 
by  A,  where  — 

A  =  RT  log  K  —  RT2v  log  C  (van  't  Hoff's  Isotherm). 

To  find  how  this  varies  with  temperature  (at  constant  volume) 
it  is  only  necessary  to  differentiate  with  respect  to  T,  thus— 


log  C  —  TJ(RZV  log  C) 

Now  the  arbitrarily  chosen  concentration  values  of  the  reactants 
and  resultants,  which  are  denoted  by  the  terms  27C"  are  not  a 
function  of  temperature  or  any  other  condition,  being  simply  a 
matter  of  choice.  That  is,  the  expression  — 


Further,  dividing  the  isotherm  by  T,  one  obtains  — 
~  =  R  log  K  —  RZV  log  C 


or 


THE    VAN  T  HOFF  ISOCHORE  145 

Now  the  Gibbs-Helmholtz  equation   states   that   for   any 
reversible  reaction — 


by  combining  equations  (i)  and  (2),  we  obtain  — 

dlogK_       JU^ 
<9T  RT2 

Since  +  U  can  be  written  —  Qv  where  —  -  Qw  stands  for  heat 
evolved  by  the  reaction  at  constant  volume  1  we  can  write  — 


<9T        "  RT2 

where  +  Qw  stands  for  heat  absorbed. 

This  equation,  which  is  one  of  fundamental  importance, 
holds  not  only  for  homogeneous  gaseous  reactions,  but  like- 
wise for  heterogeneous  reactions  as  well  as  for  solutions,  as  we 
shall  see  —  in  fact  in  all  cases  where  a  real  significance  can  be 
attached  to  "  K."  Now  suppose  that  in  a  certain  case  rise  of 
temperature  favours  the  production  of  reactants,  e.g.  dissocia- 
tion. Since  K  is  the  ratio  of  resultants  to  reactants  (raised  to 
certain  powers,  v\,  v'2t  vly  v2i  etc.),  an  increase  in  the  equili- 
brium concentration  of  resultants  due  to  temperature  means 

that  K  must  numerically  increase  with  temperature,  i.e.  —  -jjk  — 
must  be  a  positive  quantity.     Hence  —          must  be  a  posi- 


tive quantity,  that  is  —  U  must  be  positive.  Now  U  denotes 
decrease  in  total  energy,  so  —  U  denotes  increase  in  total 
energy,  that  is  heat  absorbed  at  constant  volume.  Hence  in 
the  case  considered  the  reaction  (dissociation)  must  be  accom- 
panied by  heat  absorption,  that  is  the  van  't  Hoff  Isochore 
is  simply  the  quantitative  form  of  the  Le  Chatelier-Braun 
principle  of  mobile  equilibrium  applied  to  the  particular  case 
of  a  chemical  equilibrium. 

1  Even  for  reactions  which  do  involve  a  volume  change,  the  heat 
equivalent  of  the  work  done  in  expansion  can  be  calculated  and  added  or 
subtracted  to  the  value  Q  to  give  Qv. 

T  P.C.  L 


146        A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

Integration  of  the  van  V  Hoff  Isochore. 
The  expression— 

dlogK^-U^+Q. 

err"     RT2     RT2 

is  not  of  much  use  as  it  stands  unless  we  can  integrate  it.  As 
a  first  approximation  which  holds  in  a  very  large  number  of 
cases  we  can  regard  Qv  (the  heat  absorbed  by  the  reaction)  as 
practically  independent  of  temperature  provided  we  are  not 
dealing  with  too  wide  a  temperature  range.  Taking  two 
temperatures,  T1  and  T2  (T2  >  Tj),  not  very  far  apart,  and 
supposing  that  the  heat  at  constant  volume  is  Qw  at  both  tem- 
peratures, then  the  corresponding  equilibrium  constants  Kx 
and  K2  are  connected  with  one  another  by  the  relation  — 


This  may  be  illustrated  by  the  following  data,  obtained  in 
connection  with  the  dissociation  of  nitrogen  peroxide  :  — 

N2O4  "g.  2NO2 

(cf.  Knox,  Physico-  Chemical   Calculations  ',  No.   166).      At   a 
temperature  27°C.  and  i  atmosphere  pressure  the  equilibrium 

C2 

constant  K=—  -^°2  is  0-0017.      At    111°  C.  the  constant  is 

CN204 

0-204.      What  is   the  heat  of  the   dissociation   per  mole  at 
constant  volume  ? 

We  assume  that  over  the  range  investigated  —  U  or  Qv 
remains  constant.  The  expression  just  obtained  then 
gives  us  :  — 


R  =  2  calories  per  mole  approx. 
j  =  27°  C.  =  300°  absolute 


300X384 


. 

(Heat  absorbed)  O'OOi;  84 

=  13,110  calories 


THE    VAN  'T  HOFF  ISOCHORE  147 

Another  illustration  (see  Knox's  Physico-Chemical  Calcula- 
tions •,  Problem  176). 

The  molecular  heat  of  combustion  of  hydrogen  is 
Q!  =  —  58,000  cals.,  the  minus  sign  denoting  heat  evolved  ; 
and  that  of  carbon  monoxide  is  Q2  =  —  68^000  cals.  What 
is  the  composition  at  equilibrium  of  the  water  gas  formed 
from  equal  volumes  of  water  vapour  and  carbon  monoxide 
(i)  at  a  temperature  Tj  =  800°  abs.,  and  (2)  at  a  temperature 
T2  =  1200°  abs.? 

The  reaction  H2O  -j-  CO  =  H2  +  CO2  is  made  up  of  two 
subsidiary  reactions  — 

(i)  H20  =  H2  +  i02 
and  (2)  J02  +  CO  =  CO2. 

The  heat  of  reaction  (i)  is  Qx  =  +  58,000  cals.,  and  of 
reaction  (2)  Q2  =  —  68,000  cals.  The  heat  effect  of  the  total 
reaction  H2O  +  CO  =  H2  +  CO2  is  therefore— 

Q  =  Q!  +  Q2  =  —  10,000  cals. 

The  equilibrium  constant  K  changes  with  temperature  accord- 
ing to  the  equation  — 


dT          RT2 

Integration  between  the  neighbouring  temperatures  T  and 
gives  — 

, 

(3) 


and  similarly  between  the  temperatures  T  and  T2 

K2      + 
(4)  log,  -£-    * 

From  problem  175  in  Knox's  book,  the  value  of  K  at  T=iooo° 
abs.  is  3-26.  Kj  for  T^  =  800°  abs.  ;  and  K2  for  T2  =  1200° 
abs.  may  therefore  be  calculated.  From  (3)  we  obtain  — 

lor  K   -  Ion  K       °'4343Q(Ti-T) 
logKt-  RTTj 

_    g  0*4343  X  10,000  X  200 

T'985  X  1000  X  800 
=  0-513  +  0-552=-  1-065 
and  therefore  Kx  =  1  1  '6 


48       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 
Similarly  from  (4)  — 


=  0-149 

and  K2  =  1*41 

If  Xi  is  the  fraction  of  the  water  vapour  transformed  at  Tj 
according  to  the  equation  H2O  +  CO  =  H2  +  CO2,  and  x2 
the  corresponding  fraction  at  T2,  then,  as  in  problem  175,  we 
obtain 

=  3-40 


and 


X2  =  °'542 

The  composition  of  the  water  gas  mixture  is,  therefore,  a 
T!  =  800°  abs.— 

387%  C02;  387  %H2;  ir3%H2O;  ii'3%CO; 
and  at  T2  =  1200°  abs.  — 

27-1%  C02;  27-1  %H2;  22-9%  H2O;  22-9%  CO. 

The  assumption  that  Qw  or  U  is  independent  of  tempera- 
ture cannot  be  regarded,  however,  as  anything  more  than  an 
approximation.  Experience  has  shown  that  U  may  be  repre- 
sented as  a  function  of  the  temperature  involving  a  series  of 
ascending  powers  of  T.  Thus  we  can  write  for  the  total 
energy  change  of  a  gaseous  reaction  at  any  temperature  the 
relation  — 


Where  U'0  denotes  the  heat  of  the  reaction  in  the  neighbour- 
hood of  absolute  zero.  (Actually  at  the  absolute  zero  the 
gaseous  state  can  have  no  physical  existence.)  Employing 
the  above  expression  for  U,  we  can  integrate  accurately  the 
van  't  HorT  Isochore  — 

d  log  K      —  U 
dT       ~  RT2 


THE    VAN  T  HOFF  ISOCHORE  149 

obtaining  the  expression — 


When  I  is  the  thermodynamically  indeterminate  integration 
constant.  Later  on,  in  discussing  Nernst's  Thermodynamic 
Theorem,  Chap.  XII.,  we  shall  see  how  this  constant  can 
actually  be  evaluated.  If  we  can  simply  consider  integration 
between  any  two  temperatures  Tx  and  T2  to  which  the  con- 
stants Kj  and  K2  respectively  correspond  the  integration 
constant  of  course  vanishes,  and  we  get — 

log  K2  -  log  K!  =^{^-~}-  ~  log  ^2-£(T2  -  Tj) 


The  significance  of  the  coefficients  a',  j3',  y'  will  be  made  clear 
when  we  become  acquainted  with  the  Nernst  Theorem. 


CHAPTER   VI 

Chemical  equilibrium  in  homogeneous  systems — Dilute  solutions — Ap- 
plicability of  the  Gas  Laws — Thermodynamic  relations  between  osmotic 
pressure  and  the  lowering  of  the  vapour  pressure,  the  rise  of  boiling 
point,  the  lowering  of  freezing  point  of  the  solvent,  and  change  in 
the  solubility  of  the  solvent  in  another  liquid — Molecular  weight  of 
dissolved  substances — Law  of  mass  action — Change  of  equilibrium 
constant  with  temperature  and  pressure. 

THE  APPLICABILITY  OF  THE  GAS  LAWS  TO  DILUTE 
SOLUTIONS. 

As  already  pointed  out  in  Part  I.  (Vol.  I.),  we  owe  to  van  't  HofT 
the  discovery  of  the  close  connection  between  the  behaviour  of 
gases  and  the  behaviour  of  substances  in  the  dissolved  state. 
The  quantitative  connection  between  the  two  only  holds  for 
dilute  solutions,  an  ideally  dilute  solution  being  one  in  which 
further  addition  of  solvent  does  not  cause  any  heat  effect  (heat 
of  dilution).  In  this  chapter  the  thermodynamic  proof  of  the 
applicability  of  the  gas  law  expression  PV  =  RT  to  the 
osmotic  pressure-concentration  relations  of  a  dissolved  sub- 
stance will  be  given.  It  is  important  to  notice  that  this  cannot 
be  done  simply  on  thermodynamic  grounds.  It  is  necessary 
to  assume  the  truth  of  the  experimental  law  discovered  by 
Henry,  namely,  that  the  concentration  of  gas  dissolved  in  a 
liquid  is  proportional  to  the  pressure  of  the  gas  at  the  given 
temperature.  As  has  been  shown  in  Part  I.,  this  law  holds  in 
a  large  number  of  cases,  i.e.  in  those  cases  in  which  the 
molecular  condition  of  the  solute  is  the  same  in  the  gaseous 
and  dissolved  states  respectively.  Henry's  Law  does  not  hold 
in  the  above  simple  form  for  substances  which  polymerise  or 
dissociate  electrolytically  when  dissolved  in  a  solvent.  This  has 
been  dealt  with  at  some  length  in  Part  I.  For  our  present 
purpose,  therefore,  we  shall  only  consider  the  case  of  a  gas 


VAN  'T  HOFF'S   OSMOTIC  LAW  151 

which  dissolves  "  normally  "  and  obeys  Henry's  Law  accurately. 
To  deduce  the  gas  law  for  the  dissolved  gas  it  is  necessary  to 
go  through  an  isothermal  reversible  cycle  and  equate  the  sum 
of  all  the  work  terms  to  zero.  To  carry  this  process  out  it  is 
necessary  to  postulate  the  existence  of  semi-permeable  mem- 
branes, it  being  a  matter  of  no  consequence  what  may  be  the 
mechanism  of  the  semi-permeability.  Naturally  we  are  only 
dealing  with  ideal  or  limiting  conditions,  none  of  which  can 
ever  be  realised  in  practice.  The  proof  that  the  osmotic 
pressure  of  a  dissolved  substance  is  quantitatively  identical 
with  the  gas  pressure  which  the  substance  would  exert  if  it 
were  in  the  gaseous  state  at  the  same  temperature  and  at  the 
same  concentration  (or  dilution  or  volume),  is  that  given  by 
van  't  Hoff  in  his  Lectures  (Part  -II.),  and  is  due  partly  to 
van  't  Hoff,  partly  to  Lord  Rayleigh,  and  partly  to  Donnan. 

The  Thernwdynamic  Proof  by  means  of  a  Reversible  Cycle. 

Consider  a  cylinder,  such  as  that  shown  in  Fig.  56,  con- 
taining a  quantity  of  gas  in  equilibrium  with  a  solution  of  the 
gas.  The  vapour  pressure  of  the  solvent 
is  supposed  to  be  negligible  as  compared  to 
that  of  the  gas.  The  solution  is  separated 
from  the  undissolved  gas  by  the  membrane 
bct  which  only  permits  the  gas  to  pass 
through  but  not  the  solvent,  say  water. 
The  walls  ab  and  cd  are  in  contact  with 
pure  solvent  and  are  permeable  to  the  sol- 
vent but  impermeable  to  the  solute  (dissolved 
gas).  The  system  being  in  equilibrium,  let 
us  suppose  that/  is  the  pressure  of  the  gas  (in  the  gas  phase), 
v  is  the  molecular  volume  or  volume  of  i  gram-mole  of  the 
gas  at  pressure  /,  and  similarly  let  P  be  the  osmotic  pressure 
and  V  the  molecular  volume  of  the  solute,  i.e.  dissolved  gas  in 
equilibrium  with  the  gas  at  /.  The  concentration  of  the  solute 

is  c  where  c  =  ,T.     Henry's  Law  states  that  c  a  /.     The  ends 
of  the  cylinder  are  closed  by  movable  impermeable  pistons. 


1  52        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

Suppose  both  pistons  to  be  moved  upwards  so  that  i  gram- 
mole  of  solute  is  transferred  from  the  liquid  solution  to  the 
gaseous  state.  That  is,  the  upper  piston  moves  through  a 
volume  z',  the  work  done  by  the  system  reversibly  (maximum 
work)  being  pv.  Simultaneously  the  external  surroundings 
push  the  lower  piston  up  through  a  volume  V  (the  solvent 
passing  out  through  ab  or  cd-}  the  level  of  liquid  remaining  of 
course  at  be).  The  work  done  upon  the  system  is  here  —  PV, 
the  negative  sign  denoting  work  done  on  the  system.  Hence 
the  net  work  done  by  the  system  is  pv  —  PV.  The  first 
process  is  now  completed. 

The  second  process  consists  in  restoring  the  gram-mole  of 
gas  to  the  solution  by  another  reversible  isothermal  path.  To 
do  so  we  first  of  all  imagine  the  gram-mole  of  gas  at  volume  v 
and  pressure  p  isolated  from  the  rest  of  the  system  by  pushing 
across  a  weightless,  frictionless  shutter  (this  process  involves 
no  work  in  the  ideal  case),  and  we  then  suppose  the  isolated 
gas  to  be  reversibly  expanded  to  an  extremely  great  dilution 
practically  infinite  volume,  namely  vx  .  The  maximum  work 
done  by  the  system  in  this  process  is  — 


v  =  RT        —  = 

J  ,    v 


RT  log  - 


This  gas  may  now  be  brought  into  contact  with  a  volume  V 
of  water,  this  process  being  reversible  in  the  limiting  case  in 
which  the  dilution  of  the  gas  is  practically  infinite,  for  under 
these  circumstances  the  water  would  not  absorb  any  of  the 
infinitely  dilute  gas,  i.e.  the  process  of  absorption  would  go 
infinitely  slowly  and  therefore  reversibly.  Suppose  the  piston 
to  be  lowered  gradually  until  the  gas,  i.e.  i  mole,  is  entirely 
dissolved  in  the  volume  V  of  solvent.  The  positive  work  done 
by  the  surroundings  is  equivalent  to  — 


•r. 


"0 

the  negative  sign  denoting  that  work  is  reckoned  as  done  by 
the  system.  During  the  compression,  however,/,  is  not  related 
to  Vi  by  the  usual  expression 


VAN  >T  HOFF'S   OSMOTIC  LAW  153 

when  R  represents  i  mole.  p±  is  really  smaller  than  this  (at 
any  stage  of  the  process  of  dissolving  the  gas)  because  a 
portion  of  the  gas  has  already  disappeared  from  the  gas 
phase  and  has  gone  into  solution.  This  part  which  has 
dissolved  will  evidently  amount  to  i  gram-mole  when  the 
pressure  exerted  by  the  piston  is  /  (for  i  mole  of  gas  in  V 
volume  of  solvent  is  the  original  concentration  of  dissolved 
gas  which  we  assumed  to  be  in  equilibrium  with  a  gas  phase 
at  pressure  /).  When  the  pressure  exerted  by  the  piston  is 
A>  (A</)  tne  amount  of  gas  which  has  gone  into  solution  must 

be  the  fraction  ^   of  i    gram-mole,  that  is  if  Henry's  Law 

regarding  the  direct  proportionality  between  quantity  of  dis- 
solved gas  and  pressure  be  taken  as  true.  The  quantity 
remaining  undissolved  in  the  gaseous  state  at  p±  is  therefore 

(  i  —  —  j,  and  this  is  the  mass  of  gas  to  which  we  must  apply 
the  gas  equation.  That  is,  at  any  stage  — 


RT 

or  =  — 


We  can  use  this  relation  between  pl  and  v±  to  integrate  the 
expression — 


f 

- 

J 


Thus— 


Neglecting  v  compared  to  v^  we  get  the  work  of  dissolving 
i  mole  of  gas  reversibly  and  isothermally  in  a  volume  V  of 
solvent  to  be — 

—  RT  log  ^2 
Now  we  have  got  the  gas  back  into  the  solvent  at  the  same 


154       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

concentration  as  that  of  the  original  solution  from  which  we 
had  taken  it  initially.  To  all  intents  and  purposes  this  is 
identical  with  bringing  the  gas  back  into  the  original  solution 
itself,  for  we  have  simply  to  bring  this  volume  V  of  solution 
into  contact  with  the  original  solution  when  they  will  mix, 
there  being  no  work  process  involved  in  this  since  both 
systems  are  one  and  the  same.  Hence  the  cycle  is  complete, 
and  by  the  Second  Law  of  Thermodynamics  the  nett  work 
done  in  an  isothermal  reversible  cycle  is  zero.  That  is,  the 
work  terms — 

pv  —  PV  +  RT  log  ^  —  RT  log  ^  =  o. 

or  pv  =  PV 

but  pv  =  RT 

.'.  PV  ==  RT 

That  is,  the  osmotic  pressure  obeys  the  generalised  gas  law, 
which  includes  Boyle's  Law,  Gay-Lussac's,  and  the  Avogadro 
Hypothesis.  It  will  be  observed  that  the  "R"  is  numerically 
identical  with  the  R  of  the  perfect  gas  equation.  That  is,  the 
osmotic  pressure  of  a  dilute  solution  is  related  to  the  molecular 
volume,  or  the  inverse  of  this  the  molecular  concentration 
quantitatively  as  the  pressure  of  a  perfect  gas  is  related  to  the 
volume.  For  substances  in  dilute  solution  R  =  1*985  calories 
per  gram-mole.  It  may  be  noted  that  in  the  particular  case 
in  which  V  =  v  the  osmotic  pressure  is  identical  with  the 
gas  pressure. 

The  deduction  just  given,  which  is  based  on  a  thermodynamic 
cycle,  and  also  on  the  validity  of  Henry's  Law,  proves  that  the 
osmotic  pressure  obeys  Boytts  and  Gay-Lussacs  Laws,  and  is  in 
agreement  with  the  Avogadro  Hypothesis. 

The  experimental  evidence  that  the  osmotic  pressure  of 
dilute  solutions  is  identical  with  gas  pressure  is  furnished  by 
the  work  of  Morse  and  Fraser  and  their  collaborators,  to 
which  reference  has  already  been  made  in  the  chapter  on 
solution  in  Part  I.  As  a  corollary  to  the  proof  which  has  just 
been  given,  it  should  be  mentioned  that  if  it  is  assumed  that 
the  osmotic  pressure  of  a  dissolved  gas  does  obey  the  gas  laws, 


VAN  T  HOFF^S   OSMOTIC  LAW  155 

then  it  is  possible  by  means  of  a  thermodynamic  cycle  to 
deduce  Henry's  Law  of  Absorption.  (Compare,  for  example, 
Sackur's  Lehrfaich  der  Thermochemie  und  Thermodynamik^ 
p.  209,  in  which  is  given  the  deduction  of  the  Distribution 
Law  of  which  Henry's  Law  is  a  particular  case.) 

The  applicability  of  Henry's  Law  has  recently  been  investi- 
gated over  a  considerable  pressure  range  by  Sander  (Zeit.fur 
Physik.  Chem.t  78,  513,  1912),  and  by  Sackur  and  Stern  (Zeit. 
fur  ElectrocJieniie^  18,  641,  1912),  the  object  being  to  see  what 
concentration  values  are  reached  both  in  the  gaseous  phase 
and  in  the  solution  before  the  simple  gas  law  breaks  down. 
Sackur  and  Stern  consider  they  have  shown  the  very  interest- 
ing fact,  that  in  the  case  of  carbon  dioxide  partly  dissolved  in 
various  organic  liquids  the  gas  law  breaks  down  for  the  gas 
phase  much  sooner  than  it  breaks  down  for  the  same  substance 
in  the  solution.  In  other  words  we  can  consider  that  a  solu- 
tion is  functioning  as  an  "  ideal  solution  "  to  higher  osmotic 
pressures,  than  is  the  case  with  the  gas  itself  in  equilibrium 
with  this  solution.  In  the  following  table  are  given  some  of 
the  results  of  Sackur  and  Stern,  on  the  absorption  coefficient ; 
and  in  order  to  be  able  to  work  up  to  considerable  concentra- 
tions without  introducing  very  high  pressures  which  might 
give  rise  to  some  experimental  difficulties,  measurements  were 
made  at  low  temperatures,  —  78°  C.  and  — 59°  C.,  since  the 
solubility  of  the  gas  is  greater  the  lower  the  temperature.  The 
distribution  coefficient  can  be  expressed  in  two  ways. 

Bunserfs  Definition. — The  coefficient  k'  denotes  the  quantity 
of  CO2  in  cubic  centimeters  at  o°  C.  and  at  the  pressure  of  the 
experiment  absorbed  by  i  gram  of  the  liquid. 

Ostwald's  Definition. — The  coefficient  k  is  the  ratio  of  the 
concentration  of  the  CO2  in  the  solution  and  in  the  gas  space. 
(This  is  obtained  from  the  k'  value  by  employing,  in  addition, 
data  for  the  density  of  the  solution.)  Both  k  and  k'  are  given 
in  the  following  table. 


156        A    SYSTEM  OF  PHYSICAL   CHEMISTRY 


ABSORPTION  COEFFICIENT  OF  CO2  GAS  IN  VARIOUS  SOLVENTS. 
I. — Temperature   —  78°  C. 


Solvents  — 

Methyl  alcohol. 

Acetone. 

Pressure  of  gas  in  mm. 

k 

k 

k 

of  mercury. 

50 

194-0 

120-5 

3II-0 

196-6 

100 

195-0 

119-6 

322-O 

198-1 

2OO 

202-9 

I2O-  I 

344-5 

201  5 

400 

221-5 

I22'2 

400-0 

208-8 

640 

— 

487-0 

215-7 

700 

26o-0 

126-8 

! 

II. — Temperature   —59°  C. 


Solvents- 

Ethyl  alcohol. 

Acetone. 

Methyl  acetate. 

Pressure  of 

gas  in  mm. 

V 

k 

V 

k 

V 

k 

of  mercury 

IOO 

40-85 

27-3 

97-8 

67-2 

94-3 

75-8 

200 

41  "oo 

27-2 

IOI'2      |      6S'O 

98H5 

77-1 

4OO 

42-35 

27-65 

106-6     \     69-2 

103  '6 

77-6 

700 

44-15 

28-1 

ir8-8 

72-8 

II2'9 

79-0 

It  will  be  seen  from  these  tables  that  Henry's  Law,  when 
expressed  in  terms  of  the  Ostwald  coefficient  /&,  holds  over  a 
much  wider  pressure  range  than  when  expressed  in  terms  of 
k'.  Up  to  200  mm.  k  remains  constant  within  the  limits  of 
experimental  error,  and  it  will  be  observed,  that  at  the  same 
time  the  magnitude  of  the  coefficient  shows  that  the  concentra- 
tion of  the  dissolved  gas  becomes  very  considerable.  Using  an 
empirical  formula,  Sackur  and  Stern  have  calculated  with  the 
aid  of  these  absorption  coefficients  the  values  of  the  osmotic 
pressure  of  the  various  solutions,  and  hence  the  concentration. 
To  indicate  the  order  of  concentration  obtained  we  may 
mention  that  the  solubility  of  CO2  at  —  78°  C.  in  methyl 


OSMOTIC  PRESSURE  AND  VAPOUR  PRESSURE     157 

alcohol  in  gram-moles  per  liter,  under  50  mm.  pressure,  is 
0-495  (i.e.  nearly  half  molar),  under  400  mm.  the  concentration 
is  4'O2  (i.e.  approx.  4  molar),  and  under  700  mm.  approx. 
7  molar.  It  must  be  remembered,  of  course,  that  in  the  case 
of  non-volatile  solutes  such  as  sugar  in  aqueous  solution  the 
simple  gas  laws  break  down  at  much  lower  concentrations.  As 
already  mentioned,  the  limit  up  to  which  a  solution  can  be 
regarded  as  accurately  ideal  is  about  ^N. 


THERMODYNAMIC  RELATIONS  BETWEEN  THE  OSMOTIC  PRES- 
SURE OF  DILUTE  SOLUTIONS  AND  OTHER  PROPERTIES 
OF  SUCH  SOLUTIONS. 

I.   The   Relation   between   Osmotic  Pressure   and  Lowering  of 
Vapour  Pressure  of  the  Solvent  due  to  the  presence  of  a  Solute. 

Incidentally  it  may  be  pointed  out  that  by  the  aid  of  the  Le 
Chatelier  and  Braun  principle  one  can  predict  that  the  vapour 
pressure  (of  the  solvent,  of  course)  over  a  solution  containing 
a  non-volatile  solute  is  less  than  the  vapour  pressure  over  the 
pure  solvent  itself  at  the  same  temperature.  Thus,  consider 
the  solvent  alone  in  equilibrium  with  its  vapour.  Dissolve 
some  non-volatile  substance  in  the  liquid.  According  to  the 
principle  the  system  tends  to  remain  in  its  former  state,  i.e.  it 
tends  to  lower  the  concentration  of  the  solute  as  far  as  possible 
since  it  will  thereby  be  approximating  to  the  state  of  pure 
liquid.  It  effects  this  by  causing  some  of  the  vapour  to  be 
condensed,  thereby  making  the  vapour  more  dilute,  and 
therefore  lowering  the  vapour  pressure. 

To  show  the  connection  between  the  lowering  of  vapour 
pressure  due  to  the  solute,  and  the  osmotic  pressure  of  the 
solution,  we  imagine  the  following  isothermal  reversible  cycle 
carried  out.  In  one  vessel  there  is  a  solvent  in  contact 
with  its  own  vapour  at  pressure  /0  (Fig.  57).  Vapour  can 
be  removed  by  means  of  the  piston.  In  a  second  vessel 
there  is  a  solution  of  a  non-volatile  solute,  the  osmotic  pressure 
being  P,  and  the  vapour  pressure  /  when  p  <  /0.  Pure 
solvent  can  be  removed  from  the  second  vessel  by  pressing 


158        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 


1 

1 

Vapour 
Po 

Vapour 
P 

Solvent 

Solution 
P 

Solvent 

E. 

I 

in  the  lower  piston,  which  is  fitted  with  a  semi-permeable 
membrane,  i.e.  one  allowing  solvent  to  pass  through  but  not 
solute.  The  temperature  of  both  vessels  is  the  same.  The 
cycle  consists  in  evaporating  a  cer- 
tain quantity  of  solvent  from  the 
first  vessel,  and  adding  it  to  the 
second  by  the  upper  piston,  and  re- 
moving it  again  by  the  lower  piston, 
and  finally  transferring  it  to  the  first 
vessel  once  more. 

The     expression     for     the     iso- 
thermal  reversible  transfer  of  given 
mass  of  solvent   from   vessel   I.  to 
vessel    II.   by    distillation    involves 
the    familiar    three-stage   work    process,    and    is    given     by 

[P  [Po 

the  term  —  /    vdp^  or  /     vdp.      Supposing  the  vapour  obeys 

J  PO  * p 

the  gas  laws,  then  the  work  done  by  the  system  in  the  transfer 

of  i  mole  is  RT  log  — .  If  the  quantity  is^only  dx  moles,  then 
the  work  done  is — 

dx  RT  log  ^ 

The  quantity  dx  is  now  in  the  solution.  By  using  the  lower 
piston  of  vessel  II.  one  can  squeeze  out  a  volume  dv  of  solvent 
corresponding  to  the  mass  dx  reversibly,  the  work  done  upon 
the  system  being  P</z>,  where  P  is  the  osmotic  pressure  against 
which  the  piston  was  moved.  For  such  a  small  volume  change 
as  dv  the  value  of  the  osmotic  pressure  remains  sensibly 
constant.  The  isolated  volume  dv  is  now  added  without  work 
of  any  kind  to  the  first  vessel,  and  the  initial  condition  of 
things  is  once  more  restored.  Since  the  process  has  been 
carried  out  isothermally  and  reversibly,  the  Second  Law  of 
Thermodynamics  states  that  the  sum  of  all  the  work  terms  is 
zero.  Reckoning  work  done  by  the  system  as  positive  and 
work  done  on  the  system  as  negative,  the  Second  Law  leads  to 
the  relation — 


CHANGE  IN  SOLUBILITY  159 


or 

Now  m—  =  density  of  the  liquid  solvent  =  />,  where  ;;/  is 
the  molecular  weight  of  the  solvent  as  vapour. 

Pa       P^ 

' 


which  is  the  relation  required. 

In  a  dilute  solution  P  =  RTY,  where  c  is  the  molar  concen- 
tration of  the  solution,  and  therefore  — 


. 

which  is  the  accurate  form  of  Raoult's  Law  based  on  thermo- 
dynamics.    If  the  difference  between  /  and  /0  is  small,  one 

may   substitute  in  place  of  log  —  ,  and  hence  obtain  — 

/o  P 


PQ 

which  is  the  approximate  relation  between  P  and  (/0  — P) 
already  deduced  in  Part  I.  (Vol.  I.)  on  kinetic  grounds.  Some 
numerical  illustrations  are  given  there. 

II. — The  lowering  of  Solubility  of  one  Liquid  in  another  owing 
to  the  Presence  of  a  Solute  in  one  of  the  Liquids. 

Take  the  case  of  a  system  consisting  of  water  and  ether, 
the  lower  layer  being  a  solution  of  ether  in  water.  If  one  adds 
to  the  upper  ether  layer  a  substance  soluble  only  in  ether  and 
not  in  water,  it  will  be  found  that  the  presence  of  the  solute  in 
the  upper  layer  has  caused  the  concentration  or  solubility  of 
the  ether  in  the  water  layer  to  decrease.  The  behaviour  is 
quite  analogous  to  the  lowering  of  vapour  pressure  discussed 
in  the  previous  section,  if  we  imagine  the  water  layer  to  take 


160        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

the  place  of  the  vapour  space  and  the  concentration  of  the 
ether  in  the  water  to  correspond  to  the  vapour  pressure  of  the 
solvent  in  the  previous  case.  We  can  thus  directly  apply 
the  thermodynamically  deduced  formula — 


to  the  present  case  if  we  use  s  for  p  and  ^0  f°r  /o>  where  s  is 
the  solubility  of  the  ether  in  the  water  layer  when  the  solute  is 
present  in  the  ether  layer  ;  SQ  the  solubility  of  the  ether  in  the 
water  layer  before  the  solute  has  been  added.  That  is  — 


where  P  is  the  osmotic  pressure  of  the  solute  in  the  ether 
layer,  p  the  density  of  liquid  ether  and  R  and  T  have  the  usual 
significance. 

HI.  —  Relation  between  the  Osmotic  Pressure  P  of  the  Solute  and 
the  rise  of  Boiling  Point  AT  of  the  Solvent  due  to  the 
presence  of  the  (Nonvolatile)  Solute. 

This  relation  can  be  obtained  by  means  of  a  cycle  which 
differs,  however,  from  that  employed  in  I.,  in  that  the  cycle  is 

not  carried  throughout  at  the  same 
temperature.  Consider  the  two 
vessels  figured  (Fig.  58).  That 
containing  the  solution  (vessel  I.) 
is  at  temperature  T2,  at  which 
temperature  the  solution  boils,  i.e. 
the  vapour  pressure  is  i  atmo- 
sphere. The  vessel  II.  contains  sol- 
vent also  boiling,  i.e.  vapour  pres- 
sure is  equal  to  i  atmosphere  and 
the  temperature  is  Tj.  Tj  is  less 


p,=latmos 

p=Iatmos. 

Vapour 

T2          Te 

Vapour 

Solution 
P 

Solvent 

hr" 

wperature 
I 

mperature 

n. 

FIG.  58. 


than   T9.      We    shall   denote    the 


difference  between  them  by  AT. 
The  cycle  is  as  follows  :— 
(i)  Starting  with  vessel  I.,  let  us  suppose  a  quantity  dx  of 


OSMOTIC  PRESSURE   AND   BOILING   POINT      161 

solvent  is  vaporised  from  the  solution  at  temperature  T2.  The 
heat  which  is  taken  in  from  the  surroundings  is  the  latent  heat. 
If  L  is  the  latent  heat  per  gram  of  solvent,  the  heat  taken  in 
for  mass  dx  is  ~Ldx.  At  the  same  time  the  reversible  work  pdv 
is  done  by  the  system. 

(2)  The  quantity  dx  of  vapour  is  now  cooled  to  Tlf  i.e. 
through  an  interval  AT.  If  k  is  the  specific  heat  of  the 
vapour,  the  amount  of  heat  given  out  is  a  very  small  quantity 


(3)  The   vapour   is   now  condensed  in  vessel  II.  at    T1 
reversibly,  the  work  done  upon  the  system  being  very  nearly 
—pdv,  so   that   the  work   terms   expressed   in   (i)    and    (3) 
balance  one  another.     Heat  is  given  out  in  an  amount  some- 
what less  than  l^dx. 

(4)  The  quantity  dx  now  in  the  form  of  liquid  is  separated 
without  work  from  the  vessel  II.  by  sliding  a  shutter  across. 
The  small  mass  of  liquid  is  raised  to  the  temperature  T2.    The 
heat  added  in  doing  this  is  fc'dxkT,  where  k'  is  the  specific 
heat  of  the  liquid.     Although  in  general  k  and  k'  differ  con- 
siderably, the  heat  quantities  in  the  two  cases  (Stages  (2)  and 
(4))  are  so  small  that  they  might  be  neglected,  when  compared 
to  Ldx,  particularly  since  they  are  opposite  in  sign. 

(5)  Finally,  at  temperature  T2,  add  the  mass  dx  (which 
occupies  the  volume  dV)  of  liquid  solvent  to   the   solution 
reversibly,  the  osmotic  work  being  PdV.     This  is  the  only 
nett  work  of  the  process.     The  cycle  is  now  completed.     For 
the  sake  of  making  the  cycle  clear  all  the  steps  have  been 
described  in  detail.      For  the  purpose  of  using  the  Second 

Law  expression,  dA.  =  Q— • ,  where  dA  is  the  external  work 

done,  and  Q  the  heat  taken  in  at  the  high  temperature  T,  it  is 
only  necessary  to  take  into  account  the  nett  external  work 
done,  namely  PdV  in  the  above  cycle,  and  connect  it  with 
the  Q  which  is  evidently  represented  by  the  term  ~Ldx.  The 
Second  Law  expression  in  the  present  instance  becomes 
therefore — 

PrfV  =  (Ldx)^- 

T.C.P. — II.  M 


1  62        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

Putting  — 

dx 


where  p  is  the  density  of  the  solvent,  we  obtain  — 

LpAT 

T2 

This  expression  connects  the  osmotic  pressure  of  the  solution 
with  the  rise  of  boiling  point  AT.  Since  the  solution  is  dilute, 
we  can  write  in  general  — 

P  =  RTC 

which  for*temperature  T2  becomes  P  =  RT2C,  where  C  is  the 
concentration  of  the  solute. 


This  is  the  theoretical  basis  for  the  Raoult  generalisation. 

IV.  —  The  Relation  between  Osmotic  Pressure  P  and  the  lowering 
of  Freezing  Point  AT  of  the  Solvent  due  to  the  presence  of 
the  Solute. 

We  imagine  a  cycle  very  similar  to  the  previous  one 
carried  out.  Consider  two  vessels  (Fig.  59)  I.  and  II.  Vessel  I. 
contains  solvent  (water,  say)  in  equi- 
librium with  ice  at  the  freezing  point 
T0.  The  vessel  II.  contains  an 
aqueous  solution  which  is  also  in 
equilibrium  with  ice  at  the  tempera- 
ture T  lower  than  T0  by  the  amount 
AT.  The  osmotic  pressure  in  II. 
is  P. 

ist  Stage.  —  Suppose  a  very  small 
quantity  Ax  grams  of  ice  is  with- 
drawn from  vessel  I.  at  T0.  No 
work  is  done  in  this  process,  equi- 
librium being  established  throughout. 
Now  suppose  the  ice  is  melted.  If  L  is  the  latent  heat  of  fusion 


\ 

.      |P. 

Water 

Solution 
P 

Ice 

Ice 

TO 
Vesse/I 

TO>T  [ 

T 
/esse/ff. 

FIG.  59. 

OSMOTIC  PRESSURE  AND   FREEZING   POINT     163 

per  gram  of  ice,  the  quantity  of  heat  taken  in  by  the  system 
from  the  surrounding  at  T0  is  Ldx.  In  this  process  there  is  a 
very  small  amount  of  work  done  upon  the  system  owing  to  the 
change  in  volume  of  dx  on  fusion.  This,  as  will  be  seen 
in  (3),  is  practically  neutralised  by  a  similar  work  term  in  the 
opposite  sense  at  temperature  T. 

2nd  Stage.  —  Suppose  the  mass  dx  of  liquid  water  at  T0 
approximately  occupies  a  volume  dV.  Cool  this  from  T0 
to  T.  If  k  is  the  specific  heat  of  water,  the  heat  given  out 
(a  small  quantity)  is  kdx^Y.  The  change  in  the  volume  of  the 
liquid  due  to  the  temperature  change  may  be  neglected.  The 
volume  dV  of  water  is  now  by  means  of  a  semipermeable 
membrane,  reversibly  and  isothermically  added  to  the  solution 
at  the  temperature  T.  The  maximum  work  done  by  the 
system  is  PdV. 

yd  Stage.  —  Suppose  a  quantity  doc  of  water  is  frozen  out 
of  the  solution  in  vessel  II.  A  quantity  of  heat  is  given  out 
somewhat  less  than  that  taken  in  at  the  fusion  process  in 
vessel  I.  The  system  at  the  same  time  expands  in  the  solidi- 
fication, i.e.  does  work  against  the  surroundings,  and  this  is 
balanced  by  the  small  work  term  referred  to  in  Stage  i. 

4/#  Stage.  —  The  mass  dx  of  ice  at  T  is  isolated  without 
work  from  vessel  II.,  and  is  raised  in  temperature  to  T0. 
If  k'  is  the  specific  heat  of  ice,  the  heat  absorbed  is  k'dxASY. 
This  is  a  small  quantity,  and  is  practically  balanced  by  the 
opposite  term  in  Stage  2.  The  quantity  dx  of  ice  at  temperature 
T0  is  now  added  without  work  to  vessel  I.,  and  the  cycle  is 
completed.  Applying  the  Second  Law,  we  get  — 


or 

Putting— 

dx 


the  density  of  the  liquid  water,  we  have  the  equation  — 

LpAT 
TO'« 


1  64        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

If  the  solution  is  dilute,  and  therefore  obeys  the  Gas 
Law  — 

P  =  RT0C 
LpAT 

C=W 

This  expression  has  exactly  the  same  form  as  that  obtained 
for  the  rise  of  boiling-point.  It  is  the  thermodynamical  basis 
of  the  Raoult  generalisation  regarding  the  concentration  of  a 
solution  and  the  lowering  of  freezing-point. 

Note  that  AT  is  a  positive  term  in  the  case  of  the  effect  on 
the  boiling-point,  but  a  negative  term  in  the  case  of  the  effect 
on  the  freezing-point. 

METHODS   FOR  DETERMINING  MOLECULAR  WEIGHT  OF  DIS- 
SOLVED SUBSTANCES. 

The  relations  (I.-IV.),  as  well  as  direct  measurements  of 
osmotic  pressure,  can  be  employed  to  obtain  the  molecular 
weight  of  a  given  dissolved  substance. 

Thus,  suppose  that  in  a  volume  V  of  solvent  we  dissolve 
x  grams  of  solute.  If  M  in  grams  is  the  molecular  weight  of 

oc 
the   solute  in  the  dissolved  state,       is  the  number  of  gram- 

moles  of  solute  present  in  volume  V.     The  concentration  in 

oc 
moles  per  cubic  centimetre  is  thus        -'     Now  applying  the 

Gas  Law  to  the  solution  — 

P  =  RTC 


RT* 

=          or    M  =  =r 


The  First  Method  of  obtaining  the  molecular  weight  of  a 
solute  is  to  measure  directly,  as  Pfeifer  did,  the  osmotic 
pressure  P  when  a  mass  x  is  dissolved  in  a  volume  V  at 
temperature  T.  The  numerical  value  to  be  assigned  to  R 
depends  on  the  units  used.  Thus  if  PV  be  represented  in 
calories,  R  will  be  1-98.  If,  as  is  more  usual,  P  be  measured 
in  atmospheres  and  V  in  liters  (PV  in  liter-atmospheres),  then 


MOLAR  WEIGHT  OF  DISSOLVED  SUBSTANCES    165 

R  has  the   value   0*0821.     When   P   is   known,    M   can   be 
calculated  by  the  above  relation. 

The  Second  Method  is  based  on  the  lowering  of  the  vapour 
pressure  by  the  non-volatile  solute.  The  formula  deduced 
is — 


,*      pRT 

where  m  —  molecular  weight  of  solvent. 

RT* 


Substituting 


p  — 


MV 


xm 


. 

we  have  the  equation     M  =  —  log  —  . 

pV         / 

The  determination  of  the  molecular  weight  of  solutes  by 
the  lowering  of  vapour  pressure  could  scarcely  be  regarded 
as  one  of  the  most  convenient  methods  until  quite  recently 
A.  W.  C.  Menzies  (ZeitscJt.  fur  Phys.  Chem.,  76,  231,  1911) 
has  described  a  very  convenient  form  of  apparatus  in  which 
the  lowering  of  pressure  is  measured  by  a  column  of  the 
solution  itself,  and  which  allows  accurate  determinations  of 
molecular  weights  being  carried  out  rapidly.  The  following 
table  is  an  illustration  of  the  accuracy  obtainable. 

Solvent  :    Water.       Solute  :  Sodium  Chloride. 


Mass  of  substance 
in  grams. 

Volume  of 
solution  in  c.c.'s. 

(Po  —  P)  m  mm- 
of  solution. 

Molecular  weight 
found. 

0-351 
o'579 
0-488 

38-0 
40-4 

33'8 

56-2 

84-4 
84-4 

32'5 
33'3 

33*4 

Solvent :  Benzene.     Solute  :  Napthalene. 


0-3115 

48-1                   60*  I 

128-6 

0*5118 

49-1 

105-2 

127-5 

0-3237 

46*1 

65*2 

128-0 

0*5092 

46-8 

101-6 

127-3 

Theoretical 

128*0 

1  66       A   SYSTEM   OF  PHYSICAL   CHEMISTRY 

Menzies  draws  the  following  comparison  between  this 
method  and  the  boiljng-point  rise  method  :  — 

"  The  present  method  is  capable  of  greater  accuracy,  since 
for  a  pressure  lowering  amounting  to  about  40  mm.  the  change 
in  boiling-point  is  only  o'i°." 

The  Third  Method  of  obtaining  the  molecular  weight  of 
dissolved  substances  is  that  depending  on  the  lowering  of 
solubility.  In  the  formula  — 

s        Pm 


we  substitute  the  value  of  P  in  terms  of  RT,  MV,  and  x.  None 
of  these  methods  has  come  into  such  universal  use  as  those 
depending  on  the  rise  of  boiling  point  and  the  lowering  of 
freezing  point  due  to  the  classic  work  of  Beckmann. 

The    Fourth    Method.  —  The    lowering    of  freezing  point. 


The  expression  P  =  —  ~^~ 


may  be  transformed  by  writing  P  =  RT0r-~  into  — 


M- 
" 


VLpAT 

It  will  be  observed  that  for  a  given  solvent  the  expression 

RT2 

^—^  is  a  constant  (Raoult's  constant). 
Lp 

Putting  this  equal  to  /£,  we  obtain — 

7  ^ 

Van  't  Hoff  was  the  first  to  calculate  k  on  the  above  thermo- 
dynamic  basis.     For  the  case  of  water — 

L  =  latent  heat  of  fusion  =  80  calories  per  gram. 

R  =  1*985  calories  per  degree. 

p  =  i  (approx.)  ; 


Lp 

For  the  special  case  in  which  x  is  chosen  equal  to  M  and 


MOLAR  WEIGHT  OF  DISSOLVED  SUBSTANCES     167 

V=i  c.c.,  it  is  evident  that  AT  =1863°.  This  is  the  so- 
called  "  Molecular  depression  of  freezing  point."  For  i  mole 
dissolved  in  i  liter  of  water,  the  lowering  of  freezing  point  is 
evidently  1*863°  C.  This  agrees  excellently  with  that  found 
by  Raoult.  Note,  however,  that  Raoult's  definition  of  mole- 
cular lowering  refers  to  i  mole  dissolved  in  i  gram  of  solvent, 
and  similarly  for  the  molecular  rise  of  boiling  point. 

The  Fifth  Method,  depending  on  the  rise  of  boiling  point,  is 
identical  in  principle  with  the  preceding.  Take  as  an  illustra- 
tion the  case  of  water  as  solvent  — 

L  =  latent  heat  of  vaporisation  per  gram  =  540  calories 

p  =  i  (approx.) 

R=  1-985 


Hence  for  i  mole  dissolved  in  i  c.c.,  "the  molecular 
elevation  of  the  boiling  point  of  water  "  is  5i4'8°.  For  i  mole 
dissolved  in  i  liter  of  water  the  rise  of  boiling-point  is  0*5148°. 
It  should  be  pointed  out  that  the  expressions  for  "  molecular 
depression  of  freezing  point  "  and  "  molecular  elevation  of 
boiling  point  "  are  never  realised  in  practice,  i.e.  they  have 
theoretical  significance  only.  The  solubility  of  substances 
would  not  allow  of  i  mole  of  substance  being  dissolved  in  i  c.c., 
and  even  if  this  could  be  obtained,  the  solution  would  be  so 
extremely  concentrated  that  the  solute  would  certainly  not 
obey  the  gas  law,  and  hence  it  would  not  be  justifiable  to 
write  P  =  RTC. 

A  Sixth  Method,  namely,  that  depending  upon  variation  of 
the  solubility  of  the  solute  with  temperature,  will  be  referred 
to  later. 

DETERMINATION  OF  ELECTROLYTIC  DISSOCIATION  AND  DE- 
GREE OF  HYDROLYSIS  BY  MEASUREMENTS  OF  FREEZING 
POINT  AND  BOILING  POINT. 

Since  the  methods  for  determination  of  molecular  weights 
give  in  the  first  place  the  number  of  individuals  present  in 


1 68        A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

a  solution,  it  is  clear  that  the  electrolytic  dissociation  of 
an  electrolyte  in  water  will  be  made  evident  by  abnormal 
lowering  of  freezing  point  and  rise  of  boiling  point  owing 
to  the  extra  number  of  new  individuals,  ions,  produced. 
In  fact,  a  binary  electrolyte  whose  degree  of  dissociation 
at  the  given  dilution  is  a  will  produce  the  abnormal  effect 

indicated,    the    magnitude    of    which    is  -      — .      Similarly, 

hydrolysis,  since  it  produces  new  individuals  in  solution,  will 
likewise  cause  a  further  abnormal  lowering  of  freezing  point 
and  rise  of  boiling  point.  Determinations  of  electrolytic  dis- 
sociation by  these  methods  have  been  made  by  Jahn  and 
Abegg.  Reference  may  be  made  to  a  recent  investigation  by 
W.  A.  Roth  (Zeitsch.fi/rp/tysik.  Chem.,  79,  610,  1912)  on  the 
dissociation  of  some  strong  electrolytes,  caesium  nitrate, 
potassium  nitrate,  sodium  nitrate,  and  silver  nitrate  in  water, 
by  the  lowering  of  freezing  point  method.  He  found  that 
the  Ostwald's  Dilution  Law  was  not  obeyed  by  these  salts 
(cf.  Washburn,  Chap.  V.,  Part  I.  (Vol.  I.)).  In  general,  how- 
ever, these  methods  are  not  suitable  for  this  type  of  measure- 
ment (unless  extreme  precautions  are  taken),  the  most  generally 
applicable  methods  being  those  already  given  in  Part  I.  (Vol.  I.), 
and  the  electro-metric  method,  which  will  be  given  later. 

Illustrations. — Problem  72,  in  Knox's  Physico-Chemical 
Calculations:  The  freezing  point  of  a  solution  of  0  =  0*684 
grams  of  cane  sugar  in  100  grams  of  water  is  t±  =  —  0*037°  C. ; 
and  that  of  a  solution  of  b  =  0*585  grams  of  sodium  chloride 
in  100  grams  of  water  is  /2  =  —  °'342°  C.  What  is  the 
apparent  molecular  weight  of  the  salt  at  this  concentration, 
and  what  is  its  per  cent,  dissociation  ? 

According  to  Raoult's  Law  (just  proved)— 

—  AT  =  kC  when  C  is  expressed  in  moles ; 

if  M  =  molecular  weight  of  sugar  in  solution, 

and       Ma  =  molecular  weight  of  sodium  chloride  in  solution, 

then  in  the  first  case  the  molecular  concentration  per  liter  is 

aio  .  bio 

— -,  and  in  the  second  case  -=-:- 

M  M 


MOLAR  WEIGHT  OF  DISSOLVED  SUBSTANCES     169 
Hence  by  Raoult's  Law  — 


rtIO 


342  X  0-585  >  :  0-037  = 
0-342  X  0-684 

where  342  is  the  molecular  weight  of  dissolved  sugar. 

Now  if  the  fraction  dissociated  per  mole  of  sodium  chloride 
is  a,  then  there  are  (i  —  a)  undissolved  moles  and  2a  dissolved 
ions.  Hence  total  number  of  individuals  in  solution  is  (i  -f-  a). 
If  there  had  been  no  dissociation  there  would  be  only  one 
individual  in  solution,  and  its  molecular  weight  would  have 
been  23  -f-  35*5  =  58-5.  Now  when  a  given  mass  in  grams  is 
dissolved,  its  weight  is  evidently  equal  to  the  number  of  in- 
dividuals X  weight  of  each  individual.  The  weight  of  each 
individual  is  the  apparent  weight  =  3i'65.  Hence  the  mass 
of  salt  in  grams  =  (i  -f-  0)31*65. 

If  no  dissociation  had  occurred  the  same  mass  of  salt  in 
grams  would  =  i  X  58-5 

.-.58-5  -31-65(1  +  a) 

.*.  a  =  0*85 
or  a  =  85  per  cent. 

Problem  74,  Knox's  Phy  sico-  Chemical  Calculations  :  Ether 
boils  at  35°  C.  under  760  mm.  pressure.  On  dissolving 
12-8  grams  napthalene  (M  =  128)  in  100  grams  of  ether  the 
rise  in  boiling  point  is  2-1°.  What  is  the  latent  heat  of 
vaporisation  of  ether  ? 

The  molecular  weight  of  napthalene  is  128.     Hence  the 

solution  is  --  —=  -  =  i  molar  weight  normal,  or  y^th  of 


a  mole  in  i  gram  of  ether,  i.e.  i  gram  mole  per  1000  grams  of 
ether  =  —  = 


p        1000 
Van  't  HofFs  expression  gives — 
RT2C 


ATp 

2  X  (273  +  35)2  X  i  _  g       (  calories  per  gram 
2-1  X  1000  97)         of  solvent 


1  70       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

It  must  be  remembered  that  the  solutes  considered  in  the  five 
methods  are  non-volatile. 

THE  LAW  OF  MASS  ACTION  is  APPLICABLE  TO  DILUTE 
SOLUTION. 

As  we  have  already  shown  that  the  osmotic  pressure  of  dis- 
solved substances  in  dilute  solution  obeys  the  gas  laws,  one 
can  infer  that  the  thermodynamic  deduction,  already  given, 
of  the  mass  law  for  the  gaseous  state  can  at  once  be  trans- 
ferred to  the  state  of  solution.  An  analogous  cycle  could  be 
carried  out  with  two  "  equilibrium  boxes  "  immersed  in  a 
reservoir  of  solvent  of  infinite  size  ;  a  series  of  semi-permeable 
pistons  being  required  for  the  transfer  of  the  various  sub- 
stances from  one  box  to  the  other.  Naturally,  when  the 
solution  is  of  such  a  concentration  that  the  gas  expression 

P  =  RTC  no  longer   holds,  the  work  expression  RT  log  - 

^2 

will   also  be   inapplicable,  just  as   the  analogous  expression, 
RT  log  ^  for  the  gaseous  state  is  likewise  inapplicable,  if  the 

gases  do  not  obey  the  simple  law. 

'  Having  found  in  this  manner  the  thermodynamic  justifica- 
tion of  the  gas  law  in  solution,  we  can  apply  the  van  't  Hoff 
isochore,  viz.  — 

dlogK        Q, 


When  K  is  the  equilibrium  constant,  and  Q  the  heat  of  the 
reaction  taking  place  in  the  solution. 

As  an  illustration  of  the  use  of  the  isochore  in  solution, 
we  may  now  proceed  with  the  Sixth  Method  of  determining 
molecular  weight.  Consider  a  substance  not  too  soluble  in 
a  solvent,  i.e.  one  for  which  a  solution  of  maximum  concentra- 
tion, i.e.  its  solubility,  obeys  the  gas  laws.  Suppose  that  at 
a  given  temperature  Tj.  the  solubility  is  jlf  and  at  another 
temperature  T2  the  solubility  is  sz.  In  each  case  there  is 
equilibrium  between  the  solid  phase  and  the  saturated  solution. 


VAN  T  HOFF  ISOCHORE  171 

If  the  "  reaction,"  i.e.  the  process  of  solution,  simply  involves 
the  transfer  of  i  mole  from  the  one  phase  to  the  other,  we 
can  regard  the  equilibrium  finally  reached  as  conditioned  by 

an  expression  of  the  form  -J  =  Kl5  where  K  is  a  mass  law 

constant.  C  is  the  concentration  of  the  solid  in  the  solid 
phase,  i.e.  in  itself,  and  may  therefore  be  regarded  as  unity. 
Hence  at  a  given  temperature  T±  we  have  sl  =  k-^.  At 
another  temperature  T2  we  have  sz  =  k%.  Hence  instead  of 

d  log  k  d  log  s 

the  term  —-yf—  we  can  write         °    ,  and  substitute  this  in 

theisochore,  viz. — 

d  log  s Qr  i  ds Qp 

<9T      ~  °r  ~ 


Integrating  between  the  two  temperatures  Tj  and  T2  with  the 
approximate  assumption  that  Qv  is  a  constant,  we  obtain — 

( 

log  s%  —  log  si  =  -. 

If  the  molecular  weight  of  the  dissolved  substance  is  M, 
and  Xi  grams  are  dissolved  in  V  c.c.  at  T1}  then  s±  =  |^y , 

and  at  T2,  sz  =  My .     By  measuring  the  heat  of  solution  of, 

say,  i  gram,  and  measuring  the  solubility  in  grams  per  cubic 
centimeter  at  the  two  temperatures,  we  can  obtain  M.  Van 
't  Hoff,  in  his  Lectures  (vol.  II.,  p.  59)  quotes  the  following  : — 
100  parts  of  water  dissolve  2-88  and  4*22  grams  of  succinic 
acid  at  o°  and  8-5°  respectively.  Using  the  differential  equa- 
tion as  it  stands,  and  putting  ^  as  small  finite  changes  AJ 
and  AiC,  we  get — 

RT2  Ay 
Q  =  —    — -  =  6830  calories  per  mole 

The  observed  heat  of  solution  per  gram  is  55,  and  therefore 
the  molecular  weight  =  -§f-  =  124,  calculated  =  118. 


172        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

Planck's  expression  for  the  variation  of  K  with  pressure, 
namely  — 

(d  log  K\  _    -Svu 
\     dp     /T™     RT 

where  —  2  vu  represents  the  contraction  due  to  the  reaction 
taking  place,  is  also  directly  applicable  to  reactions  in  solutions. 
This  may  be  illustrated  by  some  experiments  of  Fanjung  (Zeitsch. 
physik.  Chem.,  14,  673,  1894),  who  by  means  of  conductivity 
measurements  determined  the  dissociation  constants  (the 
Ostwald  constant)  for  a  series  of  weak  acids  in  aqueous  solutions 
at  various  pressures.  It  appears  to  be  a  general  conclusion  that 
the  process  of  ionisation  is  accompanied  by  a  contraction,  or 
decrease  in  volume.  Hence  the  Le  Chatelier-Braun  principle 
(embodied  quantitatively  in  the  above  expression  of  Planck) 
predicts  that  on  increasing  the  pressure  the  degree  of  dis- 
sociation should  increase  also,  that  is  the  dissociation  constant 
should  increase.  Fanjung  found  this  to  be  the  case.  The 
term  —  2vu  in  this  case  represents  the  contraction  due  to  the 
transfer  of  i  gram-mole  from  the  unionised  to  the  ionised 
state.  For  acetic  acid  at  18°  C.  Fanjung  found  that, 


at  i  atmosphere  pressure,  and  at  260  atmospheres 

log10K2  =  5-305 

Using  the  differential  equation  as  it  stands  — 
(d  loge  K\ 

V    dp    A 

may  be  written       2-3o2(logl°  Kg  ~  logl°  KI) 

0-259 

so  that  the  contraction  per  mole 

—  0-0821  X  291  X  2-302(0-305  —  0-254) 

=    -  £jVU  =    - 

259 
=  —  0-0108 

Since  the  pressure   has  been  expressed   in  atmospheres  and 
R  in  liter-atmospheres  (0*0821),  the  contraction  is  expressed 


EFFECT  OF  PRESSURE    ON  DISSOCIATION      173 

in  liters.  Hence  the  contraction  in  c.c.  per  mole  is  io'8  c.c. 
In  the  following  table  Fanjung's  calculated  values  for  various 
acids  are  compared  with  values  observed  by  Ostwald  (from 
measurements  of  the  volume  change  on  neutralisation  of  the 
acid  with  a  strong  base). 


Acid. 

Contraction 
observed. 

Contraction 
calculated. 

7 
10 

12 

^3 

13 

II 
II 
II 

c. 
'7 
'5 
'2 
•  I 

•8 
•8 
•8 
•4 

c.c. 

8-7 
10-8 
12-4 
I3'4 
13-3 

I2'I 
II-2 
I0'3 

• 

Acetic    

Propionic     .... 

Iso-butyric  .... 
Lactic           .... 

Succinic        ......  | 
Maleic          .            .      «   : 

i 

The  agreement  is  satisfactory. 


CHAPTER  VII 

Chemical  equilibrium  in  homogeneous  systems — Dilute  solutions  (con- 
tinued)— Outlines  of  the  electrochemistry  of  dilute  solutions. 

ELECTROCHEMISTRY  OF  DILUTE  SOLUTIONS.* 
Nernsfs  Theory  of  the  "  Solution  Pressure"  of  an  Electrode. 

IF  an  electrode  of  silver  be  placed  in  a  solution  of  silver 
nitrate  there  will  be  an  electrical  potential  difference  (repre- 
sented by  P.D.)  between  the  electrode  and  the  solution. 
When  an  electrode  is  in  contact  with  a  solution  containing 
ions  of  the  same  metal  as  the  electrode  itself,  the  electrode  is 
said  to  be  reversible.  A  cell  fitted  with  a  reversible  electrode 
is  said  to  be  a  reversible  cell.  The  simplest  type  of  reversible 
cell  is  that  denoted  by  the  term  "concentration  cell."  An 
example  of  this  may  be  represented  by  the  arrangement — 


Silver 
electrode 


N 

100 


AgNO 

* 


Silver 
electrode 


1  In  a  short  chapter  in  a  book  of  this  nature  it  is  quite  futile  to  attempt 
any  comprehensive  discussion  of  electromotive  force.  Only  the  funda- 
mental notions  can  be  indicated.  The  student  is  therefore  referred  to  the 
text-books  dealing  specifically  with  Electrochemistry,  especially  Le  Blanc's 
Electrochemistry  (English  or  German  edition),  Lehfeldt's  Electrochemistry 
(in  this  series  of  text-books),  and  the  recent  work  by  Allmand,  Applied 
Electrochemistry.  This  latter  book  is  of  special  significance  in  showing 
the  scientific  application  of  pure  electrochemistry  to  industrial  problems. 
As  regards  laboratory  experimental  methods,  i.e.  the  measurement  of  e.m.f., 
one  single  principle  underlies  all,  namely,  the  use  of  a  potentiometer. 
Details  may  be  found  in  any  of  the  text.-books  on  practical  physical 
chemistry. 


ELECTRODE  POTENTIAL  DIFFERENCE       175 

Such  a  combination  yields  quite  an  appreciable  e.m.f.,  the 
electrode  in  contact  with  the  decinormal  solution  being  in 
this  case  the  positive  pole,  i.e.  current  flows  inside  the  cell 
from  right  to  left.  The  nett  e.m.f.  of  this  cell  depends  on  the 

three  single  P.D.'s,  namely,  the  P.D.,  Ag  |  AgNO3  — ;  P.D., 

AgN03^   |  AgN03^;  and  P.D.,  Ag  |  AgNO3^.     The 

P.D.  where  the  liquids  meet  (known  as  the  liquid  |  liquid 
P.D.  or  contact  P.D.)  is  in  such  a  case  very  small  compared 
to  either  of  the  P.D.'s  at  the  electrodes.  The  mechanism  of 
the  production  of  these  potential  differences  has  been  explained 
in  an  extremely  clear  manner  by  Nernst,  and  as  they  are 
essentially  dependent  upon  the  osmotic  pressure  of  the  ions, 
it  is  only  proper  to  consider  them  in  dealing  with  dilute  solu- 
tions from  the  thermodynamic  standpoint.  Consider  the  case 
of  a  metal  like  silver  in  contact  with  an  aqueous  solution  of 
silver  nitrate,  i.e.  Ag+  ions.  According  to  Nernst  (Zeit.  physik. 
Chem.,  2,  613  ;  4,  129, 1889),  all  metals  possess  a  property  which 
he  calls  solution  pressure  or  solution  tension.  In  virtue  of  this 
property  the  metal  tends  to  drive  ions  (positively  charged) 
from  itself  into  the  surrounding  solution.  If  the  solution 
happen  to  contain  these  ions  already  (say  silver  ions  in  the 
case  considered),  then  in  virtue  of  their  osmotic  pressure,  they 
will  tend  to  drive  themselves  on  to  the  surface  of  the  metal. 
The  osmotic  pressure  of  the  ions  in  solution  opposes  therefore 
the  solution  pressure  of  the  metal  itself.  If  these  two  effects 
just  balance,  there  will  be  no  transfer  of  ions  at  all,  that  is  no 
transfer  of  electricity,  and  consequently  there  will  be  no  P.D. 
between  the  electrode  and  the  solution.  But  this  would  only 
be  an  exceptional  and  particular  case.  In  general  the  two 
effects  do  not  neutralise  one  another.  In  the  case  of  zinc,  for 
example,  in  contact  with  a  zinc  salt  solution,  the  solution 
pressure  of  this  metal  is  so  great  that  Zn++  ions  always  leave 
the  electrode  and  pass  into  the  solution.  This  process  would 
appear  at  first  sight  to  be  capable  of  going  on  ad  infinitum, 
but  as  a  matter  of  fact  only  an  exceedingly  small  mass  of  the 
metal  is  thus  transferred  (a  quite  unweighable  amount),  for  as 


176       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

the  positive  ions  pass  from  the  metal  into  the  solution,  they 
leave  the  metal  negatively  charged,  and  this  makes  the  further 
expulsion  of  positive  ions  more  difficult  owing  to  the  electro- 
static attraction  of  the  negatively  charged  metal  for  the 
positively  charged  ions.  An  equilibrium  state  is  thus  instanta- 
neously reached  when  the  solution  pressure  of  the  zinc  is  just 
balanced  by  the  osmotic  pressure  of  the  ions  originally  present 
in  the  solution,  and  by  the  electrostatic  field  set  up  between 
the  metal  and  the  layer  of  positive  ions  driven  out  by  the 
metal.  In  such  a  case  there  is  a  P.D.  between  the  metal  and 
the  layer  of  ions,  that  is,  we  say  that  there  is  a  P.D.  at  the 
electrode,  and  since  the  metal,  in  this  case  zinc,  is  negatively 
charged  compared  to  the  layer,  we  say  that  the  zinc  electrode 
is  negative  compared  to  the  solution.  In  the  case  of  silver 
the  solution  pressure  has  been  shown  (cf.  later)  to  be  so  small 

N 
that  the  osmotic  pressure  of  the  silver  ions  (say  of  a  —  AgNO3 

solution)  easily  overcomes  it,  and  deposits  positive  ions  on 
the  electrode,  an  electrostatic  field  being  set  up  between  the 
metal  and  the  solution,  but  in  this  case  the  electrode  is  posi- 
tively charged  with  respect  to  the  solution,  i.e.  there  is  a  fall 
of  potential  in  going  from  the  electrode  to  the  solution.  Sup- 
pose that  in  the  case  of  silver  dipping  into  a  solution  of  silver 
nitrate,  that  the  steady  state  is  reached  (this  is  done  instanta- 
neously), the  P.D.  being  TT  (volts),  the  solution  pressure  of 
the  metal  being  denoted  by  P,  and  the  osmotic  pressure  of 
the  solution  by  p,  i.e.  the  osmotic  pressure  of  the  positive 
ions  in  the  solution.  Then  we  can  imagine  a  small  virtual 
change  in  the  system,  namely,  the  transfer  of  8n  gram  ions 
of  the  metal  (carrying  a  quantity  of  electricity  8¥  faradays) 
from  the  electrode  to  the  solution.  Since  the  system  is  in 
equilibrium  the  work  at  constant  volume  is  zero.  The  electrical 
work  is  obviously  TrSF.  There  is  likewise  work  analogous  to 
a  three-stage  distillation  or  osmotic  work  term,  also  involved 
in  bringing  the  8n  gram  ions  from  the  solution  pressure  P  to 
the  osmotic  pressure  /.  This  work  by  analogy  is — 

S;*RT  log 


ELECTRODE   POTENTIAL   DIFFERENCE       177 
By  the  principle  of  virtual  work  we  can  write  — 

77-8F  +  S;*RT  log,  ?  =  o 
/ 


or  77-SF  =  — 

P 

SF 
but  g-  =  quantity  of  electricity  associated  with  i  gram  ion  of 

the  metal  =  «F,  when  n  is  the  valency  of  the  ion  (the  number 
of  positive  charges  carried),  and  F  is  the  faraday,  i.e.  96,500 
coulombs.  Hence  we  can  write  *  — 

RT  .        P  2-0RT          P 


_2-3o3RT  / 

~SF"  logl°P 

where  2*303  is  the  factor  for  the  reduction  of  logc  to  Iog10' 

NOTE  ON  VIRTUAL  WORK.  —  It  is  essential  to  be  clear  as  to  the  sign 
of  each  virtual  work  term  in  the  correct  application  of  the  principle.  It 
is  therefore  necessary  to  be  able  to  visualise  as  far  as  possible  the  mechanism 
of  the  process.  A  work  term  must  be  reckoned  as  essentially  positive  if 
the  force  (which  does  the  work)  assists  the  motion  of  the  mass  moved,  or 
is  in  the  same  direction  as  the  motion.  When  the  force  opposes  the  motion 
which  the  mass  or  particle  is  conceived  of  as  undergoing  the  work  term  is 
negative.  The  same  statement  regarding  sign  may  be  restated  thus  :  the 
work  term  is  to  be  reckoned  as  positive  if  the  force  and  motion  make  an 
acute  angle  with  one  another,  negative  if  they  make  an  obtuse  angle.  Thus 
taking  the  case  of  a  silver  electrode  in  silver  nitrate  solution,  consider  the 
motion  of  SF  units  of  positive  electricity  from  electrode  to  solution.  Since 
the  solution  is  known  to  be  negatively  charged  compared  to  the  electrode, 
the  electric  force  assists  the  motion  of  the  positive  charge,  and  therefore  if 
TT  be  the  potential  difference  rrSF  is  a  positive  term.  [Had  the  electricity 
been  negative,  it  would  have  been  necessary  to  write  :  —  TrSF.]  Further  the 
solution  pressure  assists  the  motion  of  the  particle  in  the  direction  con- 
sidered. Its  effect  is  therefore  positive.  The  osmotic  pressure  of  the  Ag 
ions  opposes  the  motion,  and  the  resulting  work  term  is  therefore  to  be 
given  a  minus  sign.  If  these  two  forces  appear  in  one  work  expression 
(a  logarithmic  expression),  the  term  P  must  be  in  the  numerator,  /  in  the 


1  For  another  method  of  reducing  this  relation,  see  Nernst's  Theoretical 
Chemistry,  English  translation  of  the  6th  German  edition,  p.  757. 
T.P.C. — II.  N 


178        A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

denominator.     All    the    work    terms   are   then   algebraically   added   and 

equated  to  zero,  i.e.  in  the  above  case  — 

•  P 

TrSF  +  8«  RT  log-  =  o 

RT 

To  obtain  TT  in  volts  it  is  necessary  that  -^r  should  like- 

wise be  expressed  in  terms  of  the  same  units  (i.e.  the  unit  of 
energy  must  be  the  volt-coulomb  or  joule).  The  numerical 
value  for  R  must  be  therefore  8*32.  For  a  temperature  of 
i8°C.  the  expression— 

_  0-058 


so  that  TT  volts  1  8°  C.  =  ?-5?  Iog10  ^ 

at  25°  C.  7T  volts  25°  C.  =  °^°p-  logio  | 

For  silver  and  other  monovalent  metals  ;/  is  unity.  Instead 
of  using  the  pressure  terms  P  and  /,  one  may  also  substitute 
concentration  terms  C  and  c  respectively  (where  c  denotes 
throughout  the  concentration  of  metallic  ions),  and  write  — 

RT          C       RT          c        RT 
"  ~nF  bge  c  °r  ~n¥  log  C  °r  7/F  !°g  '  +  C°nStant 

The  quantity  denoted  by  C  bears  the  same  relation  to  the 
solution  pressure  P  as  c  (the  concentration  of  the  metallic 
ions  in  solution)  bears  to/  (the  osmotic  pressure  of  the  ions 
in  solution).  It  is  difficult,  indeed  impossible,  to  ascribe  any 
real  physical  significance  to  the  quantity  C,  owing  to  the 
difficulty  of  ascribing  a  really  definite  physical  significance  to 
P  itself.  Naturally  a  theoretical  expression  of  this  nature  has 
been  subjected  to  a  considerable  amount  of  criticism.  Thus 
Lehfeldt  (Phil.  Mag.,  [V]  48,  430,  1899)  has  laid  stress  on 
the  fact  that  if  the  numerical  values  for  P  are  calculated  for 
metals  (as  of  course  can  be  done,  at  least  approximately,  from 
measurements  of  the  e.m.f.  at  the  electrode  —  for  an  account 
of  the  method  of  measuring  single  P.D.'s,  reference  should 
be  made  to  some  text-book  of  Electrochemistry)^  it  is  found 


"SOLUTION  PRESSURE  "  179 

that  these  numerical  values  vary  from  the  infinitely  great  to 
the  infinitely  small.  Thus  : — 

P  for  zinc  =  9'9  X  iolB  atmospheres 

„      nickel        =  1-3  X  10°  „ 

„     palladium  =  1-5  X  io~36         j} 

Lehfeldt  says : — "  There  are  certain  obvious  difficulties  in 
the  way  of  accepting  these  numbers  as  representing  a  physical 
reality.  The  first  of  them  is  startlingly  large  ;  that,  however, 
may  not  be  a  true  difficulty.  The  third  is  so  small  as  to 
involve  the  rejection  of  the  entire  molecular  theory  of  fluids. 
If  it  is  true  that  fluids  consist  of  molecules  with  a  diameter 
of  the  order  of  magnitude  of  io~8  cm.,  then  the  production  of 
a  pressure  so  low  is  impossible ;  for  pressure  is  a  statistical 
effect  due  to  the  impact  of  numerous  molecules.  Kriiger 
(Zdt.  physik.  Chem.,  35,  18,  1900)  replies  to  Lehfeldt's  criticism, 
pointing  out  that  the  P  or  C  term  is  really  an  integration 
constant  (this,  however,  does  not  get  us  any  nearer  the  point 
regarding  the  physical  significance  of  P  or  C,  if  indeed  there 
is  any  physical  significance  to  be  attached  at  all).  In  this 
connection  reference  should  also  be  made  to  an  earlier  paper 
by  Luther  (Zeit.  physik.  Chem.  19,  1896).  For  a  further  dis- 
cussion of  this  point,  the  reader  must  consult  a  text-book  of 
Electrochemistry.  It  may  only  be  pointed  out  here,  that  the 
vagueness  respecting  the  term  P  involves  a  corresponding 
vagueness  regarding  the  "three-stage  thermodynamic  dis- 
tillation process,"  the  work  of  which  we  have  denoted  by 

RTlogJ. 

The  general  conclusion  which  has  been  come  to,  however, 
is  that  the  Nernst  method  of  treatment  of  e.m.f.  in  cells 
involves  the  greatest  advance  in  our  knowledge  of  this 
important  phenomenon.  One  or  two  simple  cases  of  the 
application  of  the  osmotic  theory  will  now  be  given. 

Concentration  Cells  ("  Concentration  Cells  with  Transport"). 

A  concentration  cell  consists  essentially  of  two  similar 
electrodes  dipping  into  solutions  of  the  same  salt,  the  solutions 


i8o        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 


being  at  different  concentrations  of  the  salt,  ^  and  r^1  at  the 
two  electrodes,  both  solutions  being  either  in  direct  contact 
or  separated  by  a  conducting  substance  of  some  kind.  The 
source  of  e.m.f.  is  to  be  found  in  the  tendency  of  the  two 
solutions  to  equalise  their  concentrations.  The  apparatus 
used  is  represented  in  diagrammatic  form  (Fig.  60)  for  the 
case  of  silver  electrodes  in  a  water  solution  of  silver  nitrate. 
This  can  be  more  conveniently  expressed  as — 


Ag 


AgN03 
'l 


AgNOs 

*2 


Ag 


rsmtfnr    To  Potentiometer  nmnro\ 


*8~* 

flecrrode 


^ 


*-Ag 
Electrode 


V^y 


r  nQ0r  C2  or 

Concent"  C  |     other  electrolyte  such   Concent 
as  satu  rated  NH4N03. 

FIG.  60. 


This  particular  type  of  cell  is  known  as  a  concentration 
cell  of  the  First  Type.  We  may  neglect  the  liquid  |  liquid 
P.D.  between  the  solutions.  This  can  be  accomplished  in 
practice  by  the  interposition  of  saturated  ammonium  nitrate 
solution  or  saturated  KC1  at  the  junction  of  the  two  liquids. 
This  is  supposed  to  have  the  property  of  nullifying  liquid 
potential  differences.  By  means  of  the  NH4NO3  device, 
Abegg  and  Gumming  (Zeit.  fur  Electrochemie,  13,  18,  1907) 


More  strictly  c±  and  c^  refer  to  concentrations  of  the  same  ion. 


CONCENTRATION  CELLS  181 

have  determined  the  e.m.f.  of  the  above  cell  for  several  con- 
centrations of  silver  nitrate.  Experiment  shows  that  the 
electrode  in  <r2  is  the  positive  pole  of  the  cell  when  r2  is  greater 
than  flm  The  nett  e.m.f.  E  of  the  cell  is  the  difference  of  the 
two  single  electrode  e.m.f.'s  e3  and  e±. 

Diagrammatically  it  may  be  represented  as  in  Fig.  61. 
The  system  in  equilibrium  is  now  supposed  to  undergo  a  small 
virtual  change,  in  which  8F  fara- 

days    of    positive    electricity    are  •* 

transported  from  left  to  right. 
The  electrical  work  is  E8F,  and 
since  the  electric  force  opposes  the 
direction  of  motion  (the  electricity 
being  of  the  positive  kind),  we 


(cz) 


e 


must  place  a  minus  sign  before 
the  electrical  work  term  in  apply- 
ing the  principle  of  virtual  work.  FlG  6l 
Along  with  this  electrical  work 
some  osmotic  work  is  done.  Since  the  solution  pressure  P 
of  the  /<f/"/-hand  electrode  assists  the  motion  of  the  positive 
electricity  when  being  moved  from  left  to  right,  and  the 
osmotic  pressure  p^  (corresponding  to  the  concentration  c± 
of  Ag+)  opposes  the  motion,  the  osmotic  work  term  will  take 

P  C 

the  form  +  8nRT  log  —  or  8/1 RT  log  - ,  where  8n  stands  as 

before  for  the  number  of  gram  ions  of  Ag+  which  carry  the 
charge  8¥.  Similarly,  the  work  of  transferring  8n  gram  ions 
of  kg  from  solution  cz  to  the  right-hand  electrode  will  be 


log        or  +  8«RT  log      ,   since   in   this   case  the 

osmotic  pressure  /2  assists  the  motion  of  the  8n  gram  ions 
from  solution  to  electrode  whilst  the  solution  pressure  P 
Dpposes.  In  such  logarithmic  expressions  the  quantity  in  the 

denominator  really  has  a  minus  sign  before,  since  log  -  is 

identical  with  log  x  —  log  y,  thereby  indicating  that  in  the 
virtual  work  process  the  term  "  x  "  assists  the  motion  while 
"y"  opposes  it.  Now  applying  the  criterion  of  virtual  work, 


182        A   SYSTEM  OF  PHYSICAL    CHEMISTRY 

viz.  that  the  algebraic  sum  of  all  the  work  terms  may  be 
equated  to  zero,  we  obtain— 

—  ESF  -f  S;/RT  log  -  +  8«RT  log  ^=  o 

ti  C 


or 


SF 


since  £-  =  the  valency  =  n 


it  follows  that 


or 


_      RT         C  .   RT  .      c« 

E="7r1°«71+^10«3 

RT 


XI     .  <To 

-T*? 


A  verification  of  this  expression  would  involve  a  verification 
of  the  two  single  portions  comprising  it,  i.e.  a  verification  of 
the  logarithmic  relation— first  deduced  by  Nernst — which 
exists  between  the  concentration  terms  and  the  e.m.f. 

Abegg  and  Gumming  (/.c.)  obtained  the  following  results  : — 


Concentration  of 
AgNOa. 

Ratio  of  the  ionic 
concentration  f2Aj 

»          cx 

Observed  voltage. 

N  t  N 

IO  "  IOO 

N    <    N 
loo  :  looo 

9-6 

0-0563  volts 

0-0580  „ 

0-0579 

Calculation  of  the  Liquid     Liquid  Potential  in  Concentration 

Cells. 

In  the  foregoing  we  have  neglected  the  P.D.  at  the 
junction  of  the  two  liquids,  i.e.  where  the  two  solutions  meet, 
or  we  have  employed  a  device  which  automatically  reduces 
it  to  negligible  dimensions.  Where  we  do  not  employ  the 
ammonium  nitrate,  or  some  such  solution  to  annul  the  P.D., 
there  are  certain  cases  in  which  it  is  inaccurate  to  neglect  it, 
e.g.  in  the  case  of  normal  alkali  |  normal  acid.  Before  showing 
how  such  a  P.D.  may  be  calculated  we  shall  take  a  simpler 
case,  namely,  a  cell  made  up  of  the  same  solute  throughout 
(AgNO3),  but  at  different  concentrations ;  just  as  in  the  case 


LIQUID   POTENTIAL   DIFFERENCE  183 

of  the  experiments  of  Abegg  and  Gumming  already  quoted, 
except  that  we  shall  make  no  attempt  to  annul  the  liquid 
potential  difference,  but  indicate  how  it  may  be  calculated 
when  certain  data  have  been  given.  If  we  have  two  silver 
nitrate  solutions  of  different  concentratious  set  up  with  silver 
electrodes,  there  are  two  ways  in  which  the  solutions  will  tend 
to  equalise  themselves,  (i)  The  ions,  -f-  and  — ,  tend  in  general 
to  diffuse  from  the  place  of  high  concentration  to  that  of  low, 
this  being  a  natural  diffusion  process  across  the  boundary.  If 
they  travel  at  different  speeds  the  excess  of  electricity,  of  the 
sign  carried  by  the  faster  moving  ion,  is  transferred  across  the 
boundary,  so  that  the  more  dilute  solution  takes  on  the  sign 
of  the  faster  moving  ion.  An  electric  P.D.  or  "  double  layer  " 
is  formed,  whose  electric  force  finally  checks  further  diffusion 
of  the  faster  ion,  and  accelerates  the  motion  of  the  slower 
ion  going  in  the  same  direction,  so  that  eventually  both  ions 
diffuse  at  the  same  speed,  and  so  there  is  no  further  separa- 
tion of  electricity.  Let  this  steady  state  be  reached  with 
P.D.  =  <?2  at  the  interface.  (This  state  is  reached  practically 
instantaneously.)  Now  (2)  in  addition  to  this  natural  diffusion 
process,  whereby  the  two  solutions  would  be  gradually  brought 
to  the  same  concentration,  there  is  another  method  of  attaining 
the  same  end,  if  the  two  solutions  form  part  of  a  cell,  as  in  Fig. 
60,  having  two  electrodes  immersed  in  them,  and  the  electrodes 
connected  externally  so  as  to  allow  current*  to  pass.  We  can 
imagine  such  a  current  passing  with  the  result  that  silver 
dissolves  off  one  electrode,  and  is  deposited  on  the  other. 
Silver  dissolves  off  the  electrode  immersed  in  the  weaker 
solution,  the  silver  of  course  dissolving  in  the  ionic  form,  while 
at  the  same  time  an  equivalent  number  of  the  silver  ions  in  the 
stronger  solution  have  deposited  themselves  on  the  electrode 
immersed  in  the  stronger  solution.  The  NO3'  ions  have  of 
course  travelled  in  the  opposite  direction  to  that  of  the  current 
inside  the  cell  (the  direction  of  current  being  always  taken  as 
the  direction  in  which  the  positive  ions  move).  Thus  the 
stronger  solution  has  become  somewhat  weaker  (due  to  the 
deposition  of  silver  and  the  migration  of  NO3  ions  out  of 
this  compartment) ;  the  weaker  solution  has  become  more 


1 84       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

concentrated  owing  to  the  dissolving  off  of  silver  from  the 
electrode  into  the  solution,  producing  metallic  ions,  these  ions 
meeting  with  their  companion  ions  NO3'  which  have  come 
from  the  other  solution.  In  this  way  by  allowing  the  cell  to 
produce  current,  the  two  solutions  tend  to  equalise  their  con- 
centrations "  by  electrolysis,"  and  when  there  is  no  longer  a 
difference  of  concentration  the  e.m.f.  of  the  cell  falls  to  zero. 
In  actual  measurements  of  e.m.f.,  say  by  the  potentiometer, 
practically  no  current  is  taken  from  the  cell,  so  that  there  is 
no  change  in  the  actual  initial  concentrations.  Under  these 
conditions  the  e.m.f.  measured  should  be  steady  and  constant 
during  the  time  taken  for  measurement.  During  this  time  of 
course  the  e.m.f.  at  the  boundary  of  the  two  solutions  is  <?2. 
We  will  now  proceed  to  calculate  this  quantity  <?2  by  imagining 
a  virtual  change  in  the  system,  namely,  the  passage  of  an 
infinitely  small  quantity  of  current  SF  faradays  through  the 
cell  from  the  weaker  concentration  to  the  stronger.  This  is 
carried  across  the  boundary  by  silver  ions  going  from  the 
weak  to  the  strong  solution,  and  simultaneously  by  N03'  ions 
going  from  the  strong  solution  to  the  weaker.  This  passage 
of  electricity  by  means  of  electrolysis  causes  electric  virtual 
work  at  the  junction,  namely,  e28F  (volt-faradays). 

In  the  case  of  silver  nitrate  solutions  experiment  has 
shown  that  the  mobility  of  the  anion,  which  is  directly  pro- 
portional to  its  velocity  v,  is  greater  than  the  mobility  of  the 
cation,  which  is  proportional  to  u.  Hence  since  the  dilute 
solution  by  the  "  natural  "  diffusion  process  across  the  liquid  | 
liquid  boundary  takes  on  the  electrical  sign  of  the  faster 
moving  ion,  it  follows  that  the  more  dilute  solution  of  AgNO3 
(fj)  is  negatively  charged  compared  to  the  more  concentrated 
solution  (r2) ;  that  is,  there  is  a  rise  of  potential  in  passing 
from  Ci  to  ^2  which  can  be  represented  diagram matically  thus — 


In.  the  case  of  two  solutions  of  HC1,  however,  the  rise  of 
potential  is  in  the  opposite  direction,  since  the  cation  (H*) 
travels  much  more  quickly  than  the  anion  (Cl'). 


LIQUID   POTENTIAL   DIFFERENCE  185 

To  return  to  the  electrical  work  term  e28F  (in  the  case  of 
two  silver  nitrate  solutions  in  contact).  From  the  standpoint 
of  virtual  work  this  expression  must  be  written  with  a  minus 
sign,  viz.  —  ^28F,  since  the  electric  force  opposes  the  direction 
of  motion  of  positive  electricity  when  the  electricity  is  con- 
sidered as  passing  from  the  weak  to  the  strong  solution. 
(Just  the  reverse  statement  is  true  in  the  case  of  two  HC1 
solutions.)  Now  we  have  to  consider  the  osmotic  work  terms 
simultaneously  involved  in  the  transfer  of  SF  faradays  of 
positive  electricity.  The  fraction  of  the  total  charge  SF  carried 

by  the  positive  ions  is  Spf^—  )  in  one  direction,  namely, 

from  weaker  solution  (^)  to  strong  (<r2),  where  u  is  the  velocity 
of  the  positive  ions,  and  v  the  velocity  of  the  negative  ions  in 

cms.  per  sec.  under  a  given  potential  gradient  (^jT^  being 

numerically  equal  to  the  transport  number  of  the  cation). 

If  the  valency  of  the  positive  ion  is  n  then  i  gram-ion  carries  ;/ 
faradays,  but  8F  (  .  J  faradays  have  been  transported  by 
the  positive  ions,  and  hence  this  has  involved  the  transfer  of 
positive  gram  ions  (Ag+).  Similarly,  if  the  anion 


-(-      - 


has  the  same  valency  n  there  must  be  -  \-^—\  8F  negative 

ions  taking  part  in  the  process,  the  direction  of  motion  of  the 
negative  ions  being  from  the  strong  solution  <:2  to  the  weaker 
ci"  ^  P\  be  tne  osmotic  pressure  of  the  solution  ^  (the 
weaker  solution),  and  /2  De  tne  osmotic  pressure  of  c%  (the 
more  concentrated  solution),  then  the  osmotic  work  of  trans- 

ferring —  (  -      -  )  cations  from/!  to/2  is— 

11    \l('   ~\       V/ 


Note  that  since  p^  assists  and  /2  opposes  the  motion  of  the 
ions  (from  c^  to  cz),  the  p±  term  appears  as  an  essentially 
positive  term,  i.e.  in  the  numerator,  whilst  /2  appears  with  a 


i86       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

negative   sign,   i.e.    in    the   denominator   of    the  logarithmic 
expression.     Similar^,  the  osmotic  work  required  to  transfer 

SF  /    u     \ 

—  I       ,      J  anions  from/2  to/j  (i.e.  in  the  opposite  direction) 


is — 


. 

n    u  -j-  v  Pi 

The  total  algebraic  sum  of  all  the  work  terms — since  the 
process  considered  is  virtual  work — may  be  equated  to  zero. 
That  is— 


or 


or 


u  —  v  RT 


v  —  u  RT 
21  +  v    n 


^1  _]_  ^L  .  _JL__  .  RT  log  O  =: 
/!         Z'  —  W   RT  /2 


V— _U.RT 

u  + v"» 


This  expression  gives  the  P.D.  at  the  liquid  liquid  interface 
in  terms  of  the  osmotic  pressures  or  concentrations  of  the 
ions,  and  the  absolute  velocities  or  the  mobilities  of  the  ions 
(in  the  case  of  a  binary  electrolyte). 

Expression  for  the  Total  E.M.F.  of  a  Concentration  Cell. 

Again  consider  the  silver  nitrate  concentration  cell  in 
which  cz  >  ^.  The  positive  pole  of  the  cell  is  the  electrode 
in  contact  with  the  solution  <r2-  That  is,  current  tends  to  flow 
inside  the  cell  in  the  direction  indicated,  since  the  purpose  of 
the  flow  of  current  is  to  equalise  the  concentrations  ^  and  <r2, 
and  this  is  evidently  effected  by  silver  dissolving  off  at  the  left- 
hand  electrode  and  depositing  on  the  right.  Suppose  the  single 
P.D.'s  as  indicated  are  e^  e%  and  e%,  the  total  e.m.f.  being  E — 
—  pole  +  pole 


Metal 
silver 

Weaker 
concentration  q 
Osmotic 

Stronger 
concentration  r2 
Osmotic 

Metal 
silver 

pressure  /a 

pressure  /2 

Direction  of  current  inside  the  cell. 


TOTAL   E.M.F.    OF  CONCENTRATION  CELL      187 


Representing   the   single   P.D.'s   graphically,  it  will   be  seen 
that  the  following  relation  holds  — 

Ag 


Ag 


(a)  (S)  (e) 

The  net  e.m.f.  observed  E  =  es  +  ^2  ~~  ei 

Consider  the  following  virtual  change  of  the  system.  Let 
a  current  of  8¥  faradays  flow  through  the  cell  from  left  to 
right.  The  nett  electrical  work  done  is  —  ESF,  the  negative 
sign  denoting  that  the  direction  of  motion  is  opposed  by  the 
force.  Adding  this  to  the  three  separate  osmotic  work 
expressions  corresponding  to  the  three  points  (0),  (b\  and  (<:), 
and  equating  to  zero,  we  obtain — 


.  RT  log     +      •    7 

n  5/i      n    n-\-v 


n 


P 


^       RT  ,       P    ,   u  —  v   RT   ,      p,    ,   RT  ,      /2 

or       E  = log— H . log^  + log^ 

n  p\      u-\-v     n         b  /2         n 

RT         /2      21  —  v  RT        /! 

= log  *-f  H —        -  log  3- 

«  A       u-\-v    n       *  p* 


log 


i-          ^) 


RT.      /2 
—  los  ;r 


Note  that  if  the  P.D.  ^2  had  acted  in  the  opposite  sense,  the 
final  expression  for  E  would  have  been  — 


211     RT         r2 
log 

//  --  v    n        °  c 


1 88       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 


The  expression 


u  V 

is  numerically  identical  with 


the  transport  number  of  the  anion  (NO3).  The  following  table 
gives  the  numerical  values  of  the  transport  numbers  of  a  few 
anions  in  the  corresponding  salts.  Temperature  i8°C. 


Concentration  of  salt  in  equivalents  per  liter. 

Salt. 

O'OI 

O'O2 

o'os 

O'l 

o'a 

o*5 

Transport  number  of  the  anions. 

Potassium  chloride     \ 

Potassium  bromide     1 
Potassium  iodide         I 

0-503 

0-503 

0-503 

— 

— 

— 

Ammonium  chloride  ) 

Sodium  bromide        .  "1 
Sodium  chloride        .  / 

0-604 

0-604 

O'6o4 

— 

— 

— 

Lithium  chloride  . 

0*670 

0-670 

0-680 

0-687 

0-697 

— 

Potassium  nitrate 

— 

— 

— 

0-497 

0-496 

0-492 

Silver  nitrate  . 

0-528 

0'528 

0-528 

0-528 

0-527 

0-519 

Potassium  hydroxide 
Hydrochloric  acid 

0-174 

0'I74 

0-174 

0-735 

0-736 

0-738 

It  will  be  observed  from  the  above  table  of  values  (experi- 
mentally determined  by  HittorPs  or  other  method)  that  in 
the  case  of  a  good  many  salts  the  transport  number  of  the 
anion  is  in  the  region  of  0-5.  That  is,  the  mobilities  of  these 
ions  (namely,  U  and  V)  are  nearly  the  same.  This  is  the 

V 
case,  for  example,  with  silver  nitrate,  in  which  TJ   i   y  is  0*528. 

2V 

Hence  TT   i   y  is  1*056  or  nearly  unity.     Hence,  in  the  case 

of  this  salt,  we  might  as  a  close  approximation  write  the  e.m.f. 
of  the  cell,  namely  E,  as  — 


This  is  the  same  thing  as  neglecting  altogether  the  liquid  | 
liquid  P.D.  ;  for  the  above  expression  is  simply  the  e.m.f.  of 
the  two  electrodes.  In  the  case  of  alkalies  and  acids,  how- 
ever, the  value  of  the  transport  number  of  the  anion  is  far 


"  NULL  "   LIQUIDS  189 

removed  from  0*5  (thus  in  the  case  of  potassium  hydroxide  the 

V 
U  -L-~y  °f  ^e  OH'  is  0735,  and   in  hydrochloric   acid    the 

V 
value  of  u  i  y  for   Cl'   is   only  0*174),  so   that   the  factor 

2V 

^j  ,   v  has  a  very  great  effect  indeed  upon  the  total  e.m.f. 

of  the  cell  in  these  cases.  That  is  to  say,  when  two  solutions, 
say  of  hydrochloric  acid,  form  part  of  a  cell,  the  liquid  |  liquid 
P.D.  cannot  be  neglected.  To  show  the  magnitude  of  such 
liquid  |  liquid  potential  differences,  take  the  case  of  a  cell 
consisting  of— 


Hg 


o'o  i  N.  potassium 

chloride,  saturated 

with  Hg2Cl2 


o- 1  oN.  potassium 

chloride,  saturated 

with  Hg2Cl2 


Hg 


The  liquid  P.D.  in  this  case  is  due  to  o'oiN.  KCl  meeting 
o'io  N.  KCl.  The  calculated  P.D.  is  0*0008  volts,  i.e. 
exceedingly  small.  Now  set  up  the  same  cell,  but  substitute 
hydrochloric  acid  for  the  potassium  chloride  in  the  two 
cases;  the  liquid  |  liquid  P.D.  now  amounts  to  0^370  volt. 

Note  011  the  Means  Employed  to  Eliminate  Liquid  \  Liquid 
Potential  Differences. 

Mention  has  already  been  made  of  the  insertion  of  a 
saturated  solution  of  ammonium  nitrate  between  the  two  solu- 
tions. This  very  probably  eliminates  the  potential  difference 
in  the  case  of  silver  nitrate  solutions,  but  it  is  by  no  means 
certain  that  it  is  of  general  application.  The  mechanism  of 
the  effect  is  quite  obscure.  Another  method,  due  to  Nernst, 
consists  in  having  the  same  electrolyte  (KNO3  or  KCl)  present 
throughout  the  entire  cell,  the  concentration  of  this  added 
electrolyte  being  much  greater  than  that  of  any  other  electrolyte 
present.  In  this  way  the  current  in  the  cell  is  carried  mainly 
by  the  added  electrolyte,  and.  we  are  simply  left  with  the 
electrode  potentials.  This  is  a  more  theoretically  sound 
method,  but  it  has  the  drawback  of  not  being  generally  appli- 
cable (KCl  could  not  be  used,  for  example,  with  AgNO3  owing 


190       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

to  the  formation  of  AgCl),  and  further  if,  say,  potassium  nitrate 
had  been  employe^  (say  i  normal)  throughout  the  silver 
nitrate  cell,  we  would  be  met  with  the  difficulty  that  the 
extent  of  the  dissociation  of  the  silver  nitrate  would  be  altered 
owing  to  the  presence  of  the  NO'3  ion  from  the  potassium 
salt,  and  the  'extent  of  this  dissociation  alteration  cannot  be 
easily  determined  (except  perhaps  by  independent  catalysis 
experiments,  which  can  only  be  employed  in  rare  cases).  We 
would  not  know  therefore  the  numerical  values  to  assign  to 
Ci  and  <r2  in  the  Nernst  expression.  Mention  should  be  made 
of  a  very  ingenious  method  adopted  by  Cohen  (Zeit.  fur 
Electrochemie,  1907)  for  obtaining  by  direct  experiment  the 
value  of  the  liquid  |  liquid  P.D.  in  the  case  of  two  solutions 
of  zinc  sulphate.  The  following  cell  was  prepared — 


Zn 


ZnSO 


ZnSO4 
'2 


Zn 


7T 


The  total  e.m.f.  is  E1?  where  — 


TT  being  the  liquid  |  liquid  potential  difference. 

The  e.m.f.  of  the  following  cell  was  then  measured  — 


Hg 


Hg2SO4  saturated 

in  ZnSO4  solution, 

concentration  c± 


Hg2SO4  saturated 

in  ZnSO4  solution, 

concentration  c 


Hg 


TT 


The  e.m.f.  is  E2,  where — 


Adding  E!  and  E2  we  get  ZTT,  and  hence  TT  itself.  [The 
negative  sign  in  the  second  case  comes  in  because  in  this 
case  the  electrode  is  reversible  with  respect  to  the  anion  SO"4, 
and  not  with  respect  to  the  Zn".] 


HELMHOLTPS  MODE  OF  CALCULATING  E.M.F.    191 

HelmJioltis  Method  of  Calculating  the  E.M.F.  of  a 

Concentration  Cell. 

This  method 1  in  point  of  time  is  earlier  than  that  of 
Nernst.  Instead  of  following  Helmholtz's  original  method  of 
calculation,  the  simpler  modification  employed  by  Sackur  in 
his  book  (Thermochemie  nnd  Thermodynamik,  p.  273)  is  here 
followed. 

It  will  be  observed  that  the  process  which  actually  takes 
place  in  a  simple  concentration  cell,  and  which  gives  rise  to 
the  e.m.f.,  is  the  tendency  of  the  two  solutions  to  become  equal 
in  concentration.  If  instead  of  transferring  solute  from  one 
solution  to  the  other  we  were  to  transfer  solvent  by  isothermal 
distillation  from  weak  to  strong,  the  same  equalisation  of  con- 
centration could  be  obviously  brought  about.  If  we  could 
evaluate  the  expression  for  this  isothermal  distillation  work, 
we  could  equate  it  to  the  electrical  work,  for  if  we  pass  from 
one  equilibrium  stage  to  another  by  any  reversible  path  what- 
soever, the  work  done  is  the  same  no  matter  what  the  particular 
path  may  have  been.  Let  us  first  of  all  consider  two  solutions, 
which  only  differ  in  concentration  (i.e.  concentration  of  solute, 
reckoned  in  moles  of  solute  per  total  moles  of  solvent  and 
solute  present)  by  dc,  the  concentration  of  the  one  solution 
being  <r,  that  of  the  other  being  c  +  dc.  The  transfer  2  of  one 
mole  of  silver  nitrate  from  the  more  concentrated  to  the  less 
concentrated  side,  which  involves  d&  electrical  work  units 
(only  dE  since  the  concentration  difference  is  only  dc\  could 
just  be  balanced  and  annulled  by  the  transfer  in  the  same  direc- 
tion, i.e.  from  strong  to  weak,  of  n  moles  of  water  ;  where  ;/  is 
the  number  of  moles  of  water  present  (in  either  solution)  per 

1  Helmholtz's  paper  referred  to  is  translated  into  English  in  the  Phil. 
Mag.  [5]  5,  348,  1878.  (Other  papers  of  Helmholtz  are  translated  in  the 
Physical  Society  Memoirs.) 

•  Note  that  the  mechanism  of  this  transport  does  not  consist  in  the 
salt  (anion  +  cation)  going  across  the  liquid  |  liquid  boundary,  but  consists 
in  some  cations  being  dissolved  off  one  of  the  electrodes  (the  one  in  the 
dilute  solution)  and  precipitated  upon  the  other  (in  the  now  concentrated 
side)  while  the  anion  goes  across  from  strong  to  weak.  The  direction  of 
current  is  from  weak  to  strong  inside  the  cell. 


192       A    SYSTEM   OF  PHYSICAL   CHEMISTRY 

mole  of  silver  nitrate.  Note  that  n  may  be  taken  as  referring 
to  either  solution,  sjnce  the  solutions  differ  only  by  an  in- 
finitesimal amount.  The  vapour  pressure  of  the  solvent  over 
the  solution  (whose  solute  concentration  is  c]  is  denoted  by  /, 
and  the  vapour  pressure  over  the  concentrated  solution  is 
denoted  by  p  —  dp.  The  work  of  isothermally  distilling  n 
moles  of  water  between  the  two  pressure  limits  is  simply 

—  vdp  or  —  #RT  —  ,  assuming  that  the  vapour  obeys  the  gas 

laws.  Hence  on  equating  the  two  work  terms  of  the  cycle  we 
obtain  — 


P 
Now  n  obviously  depends  on  the  concentration,  for  the  con- 

centration of  solute  in  a  solution  is  simply  (  ,  \  .  Hence, 
if  the  two  solutions  considered  differ  by  finite  amounts,  the 
concentration  of  solute  being  ^  and  c2,  then  c±  =  —  37- 

and  c%  =  —  -r—  •  ;  n±  being  the  number  of  moles  of  water  per 

mole  of  silver  nitrate  in  the  solution  <r1?  and  n%  the  correspond- 
ing value  of  the  water  moles  in  <r2>  we  must  integrate  the 
above  expression  between  the  limits  n±  and  «2  m  order  to  get 
the  e.m.f.  E.  That  is— 

JVE  =  E  =  -RTf*«^ 

J  nt     / 

To  do  this  we  must  know  of  some  relation  between  p  and 
;/.     This   is  given  by  the  approximate  Raotilt  Law,  namely, 
— 


p  i 


(see  Part  I.,  Vol.  I.),  where  /0  is  the  vapour  pressure  of  water 
alone.     Hence 

ndp  _     dn 
~p        n  +  i 


HELMHOLTZ^S  MODE  OF  CALCULA  TING  E.M.F.     193 
so  that — 


/nn 
^=_ 
%> 


=  RT 


since— 


x* 

c\ 

c\  = 


and     c9  = 


This  is  the  same  result  as  is  reached  by  the  osmotic 
method  for  the  case  of  a  cell  in  which  the  liquid  |  liquid  P.D. 
is  negligible. 

NOTE. — In  the  silver  |  silver  nitrate  concentration  cell,  the 
cell  is  reversible  with  respect  to  the  cation  Ag*.  Cells  can 
easily  be  set  up  reversible  with  respect  to  the  anion.  We 
have  already  considered  one  in  dealing  with  Cohen's  method 
of  measuring  the  liquid  |  liquid  P.D.  A  simple  type  of  cell 
is  represented  by — 


Hg 


.  KC1 


dilute 


KC1 .  Hg2Cl2 
strong  <r2 


Hg 


Inside  the  cell. 


RT       2U 

:-- 


where  ^  and  t2  again  refer  to  Cl'  ions  obtained  from  the  KC1 
solutions. 

Another  such  cell  is  represented  by  the  combination  — 

Inside  cell. 


Hg 


Hg2Cl2  .  ZnCl2 
dilute  solution 


ZnCl2  .  Hg2Cl2 
strong  solution 


Hg 


The  total  e.m.f.  of  this  cell  when  the  liquid  |  liquid  P.D. 
is  taken  into  account,  is 


T.P.C.  —  II. 


194        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

where  q  and  <r2  refer  to  Cl'  ions,  i.e.  the  same  expression  as 
above  except  that  the  transport  number  of  the  cation  is 
employed  instead  of  the  anion. 

Concentration  Cells  with  Single  Solutions. 

A  cell  of  this  type  has  been  realised  and  measured  by  G. 
Meyer  (Zeitsch.  fiir  physik.  Chem.,  7,  447,  1891),  in  which 
the  electrodes  are  amalgams  of  a  given  metal  at  two  different 
concentrations.  Thus,  suppose  some  zinc  is  dissolved  in 
mercury,  the  concentration  of  the  zinc  being  clt  and  another 
amalgam  is  prepared  in  which  the  zinc  is  at  concentration  *:2, 
we  can  use  these  amalgams  as  two  zinc  electrodes  (at  different 
concentrations)  dipping  into  a  solution  of  zinc  sulphate.  If  Pj 
and  P2  are  the  respective  solution  pressures  of  the  zinc  in  the 
two  cases,  and  /zn**  the  osmotic  pressure  of  the  zinc  ions  in 
solution,  then  the  e.m.f.  of  the  cell  is  given  by  E  =  e2  —  e^ 
That  is— 


Assuming  that  the  "  solution  pressure  "  of  the  zinc  is  pro- 
portional in  each  case  to  the  concentration  of  the  zinc  in  the 
amalgam,  we  can  write  — 

RT 


This  assumes  that  the  metal  is  in  the  rnonatomic  state. 
The  following  results  were  obtained  by  Meyer  in  the  case 
of  zinc  amalgams  in  contact  with  aqueous  zinc  sulphate  :  — 


fie.     !         *, 

c, 

^observed- 

^calculated. 

11-6 

0-003366 

0-00011305 

0*0419  volt 

o  *  04  1  6 

18*0 

0-003366 

0*00011305 

0-0433 

0-0425 

12*4 

0*002280 

o  *  0000608 

0-0474 

0-0445 

60*0 

0*002280 

0*0000608 

0-0520 

AMALGAM  CELLS. 


195 


For  a  further  account  of  similar  cells  and  the  questions 
which  arise  in  connection  with  them,  see  Le  Blanc,  Electro- 
chemistry',  English  edition,  p.  185.  It  may  be  pointed  out 
that  arguing  in  an  inverse  way,  one  may  employ  the  values  of 
the  e.m.f.  obtained  to  determine  the  molecular  (or  atomic 
state),  i.e.  the  molecular  weight  of  the  zinc  in  the  amalgam. 
As  regards  the  e.m.f.  of  concentration  cells  containing  solvents 
other  than  water  relatively  little  is  known,  owing  to  the  diffi- 
culty of  obtaining  reliable  values  for  the  degree  of  dissociation. 
In  acetone,  silver  nitrate  seems  to  act  in  the  way  predicted  by 
Nernst's  formula.1  For  a  short  survey  of  the  subject  the 
reader  is  referred  to  Carrara's  article  on  "The  Electro- 
chemistry of  Non-aqueous  Solutions"  in  Ahretis  Sawmlung. 


Other  Types  of  Concentration  Cells. 

Besides  cells  of  the  types  already  mentioned,  cells  can  be 
constructed  with  electrodes  giving  anions  directly  (negatively 
charged  ions).  Thus  the  following  cell,  in  which  Iodine  acts  as 
the  electrodes,  has  been  realised. 


Iodine 
lectrode 


I2  in  KI  solution, 
giving  KI3 


I2  in  KI  solution, 
giving  KI3 


"Iodine 
electrode 


In  the  actual  setting  up  of  this  cell,  it  is  sufficient  to  have 
lean  electrode  of  platinum  inserted  in  each  side,  in  contact 
nth  some  solid  iodine  at  the  bottom  of  each  vessel. 

We  may  also  have  gas  cells ,  i.e.  cells  in  which  a  gas  such  as 
hydrogen  or  oxygen  functions  as  the  electrode.  This  is  also 
realised  by  inserting  a  platinum  electrode  and  allowing  a 
stream  of  the  gas  to  bubble  through  the  solution  in  contact 
with  the  electrode,  the  bubbles  also  striking  the  piece  of 
platinum.  In  such  cases  the  platinum  is  chemically  inert— it 
simply  acts  as  a  mechanical  device  to  give  rigidity  to  the  gas 
electrode.  Thus  the  following  cell  can  yield  a  perfectly 

1  A.  P.  Roshdestwensky  and  W.  C.  Me  C.  Lewis,  Journ.  Chem.  Sec., 
99,  2138,  1911 ;  ibid.,  101,  2094,  1912. 


1 96        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

definite  e.m.f.,  the  e.m.f.  depending  on  the  logarithm  of  the 
ratio  of  the  concentwition  of  H*  ions  as  predicted  by  Nernst's 
Theory. 


Electrode  H2 

(Platinum 
Hydrogen  gas) 


Hydrochloric 
acid 


Hydrochloric    H2  electrode 
acid 


As  a  practical  point,  it  may  be  mentioned  that  the  chief 
difficulty  in  setting  up  such  gas  cells  is  due  to  the  fact  that 
the  platinum  has  already  dissolved  some  oxygen  from  the 
atmosphere,  and  this  in  contact  with  OH'  necessarily  present 
in  aqueous  solutions,  causes  an  oxygen  concentration  potential 
to  be  set  up.  It  is  necessary,  therefore,  to  remove  the  oxygen 
as  completely  as  possible.  For  details  a  text-book  on  Electro- 
chemistry must  be  consulted.  A  very  complete  list  of  various 
sorts  of  cells,  classified  under  seven  heads,  is  given  by  W.  D. 
Bancroft  (Journal  Physical  Chemistry,  12,  103,  1908). 


The  Calculation  of  Liquid  \  Liquid  Potential  Differences  between 
Solutions  containing  non-identical  Electrolytes. 

The  expression  for  the  liquid  |  liquid  P.D.,  namely— 
U  —  V  RT  .      <i 

62  =  TT-L  V  ~W   i0g  7 

U  -j-  V   nr          c% 

already  deduced,  is  only  applicable,  of  course,  to  the  case  in 
which  we  are  dealing  with  one  and  the  same  binary  salt 
(AgNO3)  at  two  different  concentrations.  We  shall  now  take 
up  the  somewhat  more  complicated  case  of  two  binary  electro- 
lytes with  different  cations  but  the  same  anion.  Thus,  suppose 
that  the  following  cell  is  set  up — 


Calomel 
electrode 


HC1  solution, 
Solution  I. 


KC1  solution, 
Solution  II. 


Calomel 
electrode 


We  shall  only  consider  the  simplest  possible  case,  namely 
that  the  solutions  of  KC1  and  HC1  are  identical  in  concentra- 
tion (<:),  and  that  the  electrolytic  dissociation  of  each  salt  is 


CONTACT  P.D.  BETWEEN  ELECTROLYTES      197 

complete,  and  also  that  the  valency  of  the  ions  is  the  same. 
This  means  that  the  chlorine  ion  has  the  same  concentration, 
c  gram  ions  per  liter,  throughout  the  cell.  There  is  a 
liquid  |  liquid  potential  difference  at  the  contact  of  solution  I. 
and  solution  II.,  owing  to  the  different  mobilities  of  the  K' 
ions  and  the  H'  ions.  We  have  to  calculate  what  this  will  be. 
In  the  first  place,  however,  it  is  necessary  to  see  what 
assumptions  are  to  be  made  regarding  the  nature  of  the 
transition  layer  between  solution  I.  and  solution  II.  There 
are  two  ways  of  looking  at  this.  First  we  may  consider  with 
Planck  l  that  the  two  solutions  have  been  so  brought  together 
that  the  boundary  is  initially  sharp.  Natural  diffusion  will, 
however,  commence  and  cause  the  sharpness  to  disappear.  If 
mixture  takes  place  by  diffusion  alone t  the  concentration  in 
any  layer  is  determined  by  two  independent  variables,  the 
rates  of  diffusion  of  the  salts.  On  the  other  hand,  we  can 
imagine,  instead  of  a  sharp  boundary  set  up  initially,  that 
a  connecting  layer  is  formed  by  causing  a  part  of  the  solu- 
tions to  be  mechanically  mixed.  This  connecting  layer  is 
really  a  series  of  mixtures  of  the  two  solutions  in  all  pro- 
portions, for  in  the  body  of  solution  I.  the  K'  ion  concentration 
is  zero,  and  in  the  body  of  solution  II.  the  H'  ion  concen- 
tration is  likewise  zero.  This  is  the  case  considered  by 
P.  Henderson,2  and  as  it  seems  to  be  more  easily  realised 
in  practice  than  Planck's  arrangement,  and  still  more  because 
it  allows  of  a  more  simple  mathematical  treatment  in  calcu- 
lating <?2>  it  is  Henderson's  method  which  will  be  followed  in 
the  present  case.  Let  us  suppose  that  in  the  cell  above 
described,  where  c  represents  the  concentration  of  Cl'  ions 
throughout,  that  the  mobility  of  the  H'  ion  is  denoted  by  «l5 
the  mobility  of  the  K*  ion  by  z/2>  and  the  mobility  of  the 
Cl'  ion  by  v.  Suppose  i  faraday  of  electricity  to  pass 
through  the  cell.  Consider  a  region  in  the  connecting  layer 
in  which  the  solution  consists  of  x  parts  of  KC1  and  (i  —  x) 
parts  of  HC1,  which  is  the  same  thing  as  saying  that  there 

1  Wied.  Ann.,  40,  561,  1890. 

2  P.  Henderson,  Zeitsch.  physik.  C/iem.,  59,  118,  1907  ;   ibid.  63,  325, 
1908. 


198        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

are  x  gram  ions  of  K'  +  (i  —  x)  gram  ions  of  H'  ions  -f  i 
gram  ion  Cl',  and  -suppose  that  i  faraday  of  electricity  is 
transferred  from  this  region  to  another  where  x  is  now  x  -{-  dxt 
and  correspondingly  i  —  x  is  now  (i  —  (x  -\-  dx)  ).  This 
takes  place  by  H'  ion  and  K'  ion  travelling  in  the  positive 
direction,  Cl'  in  the  negative.  The  fraction  of  i  gram 
equivalent  of  each  ion,  which  takes  part  in  the  transfer  of 
the  faraday  is  given  by  the  expression  — 

_  (concentration  of  the  ion)  X  (mobility  of  the  ion) 
27  (concentration)  X  (mobility) 

where  the  denominator  denotes  the  sum  of  all  terms 
involving  the  product  of  concentration  into  mobility  of  each 
ion  present.  In  the  particular  case  chosen  — 

__         H1       x)ui  __  C  fractional  number  of  gram 

H       c(i  —  X)U-L  +  cxu%  -\-cv      \      equivalents  of  H*  ion 


__  cxu<i  -  =  f  fracti°nal  number  of  gram 

C(TL  —  x)ui  +  fxuz  +  cv      \      equivalents  of  K'  ion 


_ 
C1'      c(i  —  x) 

The  osmotic  work  involved  in  this  transfer  is  represented 
by  ^A  when  —  . 

,.        p    —  RT^ 
dPi.  =  2v  -  - 

P 
where  /  is  the  osmotic  pressure. 

The  total  work  A  of  the  transfer  of  i  faraday,  when  the 
whole  layer  is  taken  into  account,  is  the  integral  of  the  above 
expression  ;  the  integration  being  carried  out  between  the 
limits,  when  x  =  o  to  x  =  i,  for  in  the  body  of  solution  I. 
x  =  o  and  in  the  body  of  solution  II.  x  =  i. 

Now,  since  osmotic  pressure  is  proportional  to  concentra- 
tion in  dilute  solutions,  we  can  write  for  the  H'  ions  :  — 

—  dp  _     dx 
p         i  —  x 

—  dp      —dx 
and  for  the  K  ions  —  = 

x 


HENDERSON'S   FORMULA  199 

The  difference  in  sign  of  dx  in  the  two  cases  is  due  to  the 
fact  that  in  any  two  contiguous  regions  in  the  layer,  if  there 
is  an  increase  in  K',  there  is  a  corresponding  decrease  in  H', 
so  that  at  any  point  the  total  concentration  of  cations  is  just 
equivalent  to  the  anion  Cl',  which  remains  at  constant  con- 
centration throughout.  The  work  term  in  connection  with  the 
anion  Cl'  is  obviously  zero,  since  its  concentration,  and  there- 
fore its  osmotic  pressure,  is  constant.  We  thus  obtain  for  the 
work  term  A  the  algebraic  sum  of  the  two  work  terms  for  H' 
and  K*  respectively,  viz.— 


A  — 


[As  a  matter  of  fact,  Planck's  formula,  to  which  reference  will 
be  made  in  a  moment,  reduces  to  this  expression  for  the 
simple  case  chosen,  viz.  identity  of  one  ion  (Cl')  in  the  two 
salts,  and  identity  of  concentration  of  the  salts.] 

The  validity  of  the  above  formula  has  been  examined 
recently  by  N.  Bjerrum  (Zeit.fiir  Electrochemie^  17,  391,  1911), 
in  which  the  experimental  arrangement  was  such  as  to  give  a 
mechanically  mixed  boundary.  Two  calomel  electrodes  were 
employed  in  the  cell,  the  solutions  in  contact  being  also 
chlorides  at  the  same  concentration  on  each  side.  The  follow- 
ing table  gives  a  few  of  Bjerrum's  data  ;  — 


200        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 


• 

^£ 

H     jo-g 

Solution  I. 

Solution  II. 

ill! 

rlil 

Sum  of 
calculated 
P.D.'s. 

Total 
E.M.F. 
of  cell 
observed. 

33 

3  ^x 

o-i,  N.  HC1 

o-i,  N.  KC1 

—  0*0016 

—  0-0276 

—  0*0292 

—  0-0278 

o*i,  N.  NaCl 
o-oi,  N.  NaCl 

0*1,  N.  KC1 
0*01,  N.  KC1 

0*0005 

0*0002 

0*0050 
0*0048 

0-0055 
0-0050 

0*0041 
0*0039 

The  agreement  of  the  last  column  is  only  moderate,  and  it 
is  difficult  to  say  how  far  this  may  be  due  to  experimental 
error,  for  the  total  e.m.f.'s  are  very  small.  Planck  (I.e.)  has 
worked  out  the  case  in  which  the  solutions  in  contact  are 
different  in  concentration,  and  are  likewise  different  in  com- 
position (i.e.  no  ions  being  necessarily  in  common),  but  the 
ions  have  all  the  same  valency.  The  method  of  deduction 
will  be  found  in  the  paper  cited ;  it  must  suffice  here  simply 
to  write  down  the  formula  obtained  for  the  liquid  |  liquid 
potential  difference. 

Planck's  formula — 

RT  . 
e2=^-^=wloge, 

where  £  is  denned  by  the  relation — 

logefi_loge£ 


where  n,  RT  and  F  have  the  usual  significance  and  u-^  and  #2 
are  the  velocities  of  the  cations,  v±  and  #2  the  velocities  of  the 
anions,  and  ^  and  *r2  tne  corresponding  ionic  concentrations. 

Planck's  formula  has  been  extended  by  K.  R.  Johnson 
{Ann.  der  Physik,  14,  995,  1904)  to  the  case  in  which  the 
valency  of  the  ions  is  not  the  same. 

Henderson  (I.e.)  has  likewise  dealt  with  the  general  case,  in 
which  the  two  solutions  differ  in  concentration,  and  have 
not  necessarily  any  ion  in  common,  and  the  ions  not  being 


PLANCK'S   FORMULA 


201 


necessarily  of  the  same  valency.  This  method  is  more 
readily  followed  than  that  of  Pianck,  but  there  is  not  space  to 
give  it  here.  The  final  expression  reached  is  :— 


' 


Where  in  the  one  solution  — 

UL  = 


Ui  =  u^n^  -f- 


etc- 

z  -f-  etc- 
etc. 


the    corresponding    quantities    for   the    second    solution    are 
denoted  by  U2,  V2l  Us,  V^. 

This    formula    may  be   illustrated    by   some  of  Bjerrum's 
data  (I.e.)  — 


c  **» 

3     >>„ 

Solution  I. 
Normality. 

Solution  II. 
Normality. 

fill 

m 

111! 

Sum  of 
calculated 
P.D.'s. 

Observed 
E.MF. 
of  cell. 

3  " 

n4         w 

o-oi  HC1 

o-ioKCl 

0*0562 

—  0   OIOI 

—  0-0461 

0-0458 

o-oi  NaCl 

.    o-ioKCl 

0-0570 

—  0-0028 

0*0542 

0*0553 

0-09  KC1) 
o-oi  HC1J 

o-io  KC] 

—  0-0002 

—  0-0050 

—  0*0052 

—  0-0041 

O-i  NaCl 

(0-09  KC1 
\o-oi  HC1 

0-0007 

0-0097 

0-0104 

0-0083 

A  critical  study  of  the  relative  applicability  of  Planck's  and 
Henderson's  formulse  has  been  undertaken  by  A.  C.  Gumming 
(Trans.  Faraday  Soc.,  8,  86,  1912;  ibid.,  9,  174,  1913)-  A 
final  conclusion  has  not  yet  been  reached,  though  the  evidence 
seems  to  point  to  the  greater  applicability  of  Henderson's 
formula.  As  a  matter  of  fact,  although  the  views  taken 
regarding  the  nature  of  the  separating  boundary  differ  widely 
in  the  two  methods  of  treatment,  the  numerical  values  of  the 
liquid  |  liquid  P.  D.  are  not  correspondingly  very  widely  dif- 
ferent. The  following  graph  (Fig.  62)  taken  from  Cumming's 


202        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

paper  (1913)  shows  the  liquid  |  liquid  P.D.  in  the  case  of 
HC1  and  KC1  solutions  of  different  concentrations  calculated 
by  Planck's  and  Henderson's  formulae  respectively.  It  will  be 
seen  that  the  differences  become  appreciable  only  when  the 
ratio  of  HC1  to  KC1  becomes  very  large. 

STANDARD  ELECTRODES  OR  STANDARD  HALF  ELEMENTS. 

Since  the  e.m.f.  of  any  cell  is  necessarily  made  up  of  at 
least  two  parts  corresponding  to  the  electrodes,  it  is  necessary, 


1000:1 


50  100  150 

MiUivo  Its 
FIG.  62. 

for  the  purposes  of  comparison  in  many  cases,  to  make  one  of 
the  electrodes  and  the  liquid  in  which  it  is  immersed  (such 
being  called  the  half  element)  possess  a  constant  value.  This 
electrode  must  be  easily  and  conveniently  set  up.  It  must 
give  under  the  same  conditions  of  concentration  and  tempera- 
ture, the  same  potential  differences,  i.e.  it  must  yield  repro- 
ducible values  in  the  hands  of  different  operators.  It  must 
likewise  have  as  small  a  temperature  coefficient  as  possible. 
One  half  element  used  frequently  for  this  purpose  is  the 
calomel  standard  electrode.  It  consists  essentially  of  some 
very  pure  mercury  acting  as  the  electrode,  external  connection 


STANDARD   ELECTRODES 


203 


being  made  by  means  of  a  platinum  wire  immersed  in  the 
mercury.  The  liquid  in  contact  consists  of  a  solution  of  KC1 
saturated  with  calomel  Hg2Cl2 ;  saturation  being  guaranteed  by 
having  a  layer  of  calomel  (mixed  with  mercury  to  ensure  the 
absence  of  mercuric  chloride  HgCl2)  placed  over  the  mercury. 
The  half  element  is  usually  made  in  the  shape  illustrated 
(Fig.  63).  When  i  normal  potassium  chloride  is  employed 
the  electrode  is  called  the  "  normal  calomel  electrode." 

N 

When  —  KC1  is  used  it  is  called  the  "  decinormal  calomel 
10 

electrode."  The  absolute  values  of  these  two  electrodes  are 
known  approximately,  that  of  the  "  normal  calomel "  being 
0*56  volt,  the  "decinormal 
calomel"  being  0-6 1  volt.  The 
mercury  in  each  case  is  posi- 
tively charged  with  respect  to 
the  solution  of  mercury  ions 
produced  from  the  calomel. 
When  used  as  a  reference  stan- 
dard, the  P.D.  of  the  "normal 
calomel"  is  set  arbitrarily  at 
zero  and  other  half  elements 
referred  to  it.  Another  standard 
electrode  consists  of  a  hydrogen 
electrode  (the  gas  being  at 


FIG.  63. 


atmospheric  pressure)  immersed  in  a  normal  solution  of  H" 
ions  (a  little  over  normal  HC1  solution).  Taking  this  as 
zero  P.D.,  the  actual  P.D.  of  the  calomel  is  +0-283  volts. 
That  is,  if  we  connect  up  a  normal  hydrogen  electrode  with 
a  normal  calomel,  the  cell  will  give  a  current,  the  current 
passing  inside  the  cell  from  the  hydrogen  electrode  to  the 
mercury.  The  mercury  is  therefore  the  positive  pole  of  the 
cell,  the  e.m.f.  of  which  is  0-283  volt.  Taking  the  hydrogen 
as  zero,  it  is  obvious  that  on  the  same  scale  the  calomel 
is  +0-283  volt.  A  list  of  such  standards  is  given  by  F. 
Auerbach  (Zeit.filr  Elektrochemie,  18,  13,  1912). 


204        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

The  Electrolytic  Potential  ("  E.P."). 

This  is  the  P.U.  at  a  reversible  electrode  (say  silver  in 
contact  with  AgNO3  solution),  when  the  concentration  of  the 
ions  is  normal.  The  expression  for  a  single  potential  of  this 
kind  is,  as  we  have  already  seen,  given  by — 

—  RT          P      —  RT          C 
^-log</=-^  log«7 

if  c  =  concentration  of  the  ions  is  unity  (in  i  normal  ionic 
solution),  log  c  —  o,  and  hence  the  electrolytic  potential  "  E.  P." 
is  given  by— - 

,£  T-.  T»  »       —  R^  i        /-       —  R1 

~^10g.C:  wj? 

This  is  the  nearest  approach  we  can  make  to  getting  a 
physical  interpretation  of  P,  i.e.  P  is  a  quantity  whose  logarithm 
is  proportional  to  the  electrolytic  potential  of  the  electrode  in 
question.  The  following  table  gives  the  electrolytic  potentials 
of  a  few  electrodes,  the  values  in  the  first  column  being  referred 
to  the  "  normal  calomel "  electrode  as  a  zero  electrode,  the 
values  of  the  second  column  referring  to  the  "  normal 
hydrogen  "  as  a  zero  electrode. 

ELECTROLYTIC  POTENTIALS. 


Electrode. 

E.P. 

"  Normal  calomel  " 
electrode  =•  o. 

E.P. 

"  Normal  hydrogen  " 
electrode  =  o. 

Platinum       .... 
Silver       

ca  +  0-580 

+  o*  <»i  s 

ca  +  0  863 
+  0-718 

Mercury  
Copper    

+  0-467 
-f-  o  *  046 

+  0-750 
+  O*  3Q2 

Hydrogen     .... 
Tin    .... 

-  0-283 

ca  —  0*4.  7  ^ 

^FO-OOO 

ca  —  o-  192 

Zinc   

—  I  'O53 

—  0-770 

Potassium  *   .      .      .      . 

-3*48 

-3*20 

Oxygen    
Iodine      

+  o-  i  10 
+  o*  34.1? 

+  0-393 
+  0-628 

1  Metals  like  potassium,  which  react  with  water,  have  to  be  determined 
indirectly.  Cf.  G.  N.  Lewis  and  Kraus,  J.  Amer.  Ckeni.  Soc.  32,  1463 
(1910),  for  measurements  in  the  case  of  sodium. 


ELECTROLYTIC  POTENTIAL  205 

The  positive  sign  means  that  on  joining  up  the  cell,  one- 
half  being  the  standard,  the  other  half  the  metal  under  investi- 
gation in  a  normal  solution  of  its  ions,  the  current  flows  inside 
the  cell,  from  the  standard  electrode  to  the  metal  electrode, 
i.e.  the  metal  electrode  is  the  positive  pole  of  the  cell.  The 
negative  sign  of  course  indicates  that  the  electrode  examined 
is  the  negative  pole  of  the  cell. 

Mechanism  of  Electrolytic  Conduction  and  Discharge  of  Ions 
on  Electrodes. 

It  might  appear  at  first  sight  that  this  would  be  of  a  simple 
character,  but  as  a  matter  of  experience,  the  question  is  by  no 
means  easy  to  deal  with  experimentally.  When  we  consider 
the  simplest  case  in  which  a  metal  electrode,  like  silver,  is  in 
contact  with  a  solution  containing  a  considerable  quantity  of 
its  ions,  say  silver  ions,  the  process  of  the  discharge  of  an  ion 
is  represented  by  the  passage  of  an  electron  from  the  electrode 
on  to  the  ion,  which  thereby  loses  its  electric  charge  and  is 
precipitated  upon  the  metal.  Thus  — 

Ag  +  -(-  ®  ->  Ag  metal 

the  electron  being  denoted  by  the  symbol  ®.  If  the  reaction 
is  one  of  ion  formation,  we  have  to  assume  that  the  reaction 
is  — 


The  presence  of  the  electron  symbol  appearing  on  one  side 
of  the  reaction  equation  denotes  a  transfer  of  electricity,  and 
the  reaction  can  only  take  place  when  current  is  allowed  to 
flow.  The  direction  of  current  is  taken  as  the  reverse  of  the 
direction  of  transfer  of  electrons,  so  that  in  the  first  case  the 
current  flowed  from  solution  to  electrode,  and  in  the  reverse 
sense  in  the  second  case. 

Electrolysis  of  a  Complex  Salt. 

Potassium  silver  cyanide  solution  is  a  very  convenient  solu- 
tion to  use  for  the  deposition  of  silver  at  the  cathode.     There 


206       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

are,  however,  very  few  actual  Ag'  ions  which  could  react  with 
the  electron  passing  ^>ut  from  the  cathode,  when  the  ion  is  in 
close  proximity  according  to  the  equation  — 

Ag+  -f  ®  ->  Ag 

According  to  Wilsmore,  what  happens  is  that  the  complex 
silver  containing  anion  —  the  salt  being  2K+  and  Ag(CN)4~ 
although  it  is  naturally  streaming  away  from  the  cathode, 
reacts  with  the  electron  at  the  cathode  thus  — 


+  2©  ->  Ag  -f  4CN 
Remember  that  this  is  the  cathode  process,  two  electrons 
passing  from  the  electrode  to  the  solution.  Naturally  the  CN 
formed  starts  to  go  towards  the  anode.  When  electrolysis  is 
prolonged  there  may  be  no  longer  sufficient  Ag(CN)4~ions  to 
so  react  (partly  because  they  have  been  used  up  as  above,  and 
partly  because  they  have  migrated  too  far  from  the  electrode 
(cathode)),  and  the  K'  begins  to  discharge,  making  the  cathode 
liquid  alkaline.  At  the  anode,  if  it  is  of  silver,  the  above 
reaction  is  just  reversed,  viz.  :  — 

Ag  +  4CN  ->  Ag(CN)i"+  2© 

the  two  electrons  being  given  up  to  the  electrode.     Probably 
also  the  complex  Ag(CN)4   can  further  react. 

Anions  may  thus  react  at  both  cathode  and  anode.  In 
these  cases  considered,  where  a  silver  electrode  either  produced 
Ag*  ions,  or  had  Ag  deposited  upon  it  from  a  solution  contain- 
ing Ag'  ions,  the  electrode  is  called  a  non-polarisable  one. 
That  is  to  say  if  we  set  up  a  silver  |  silver  nitrate  cell  consisting 
of  two  silver  electrodes  in  one  and  the  same  solution  of  silver 
nitrate,  and  cause  electrolysis  by  impressing  an  external  e.m.f. 
upon  the  cell  from  a  battery  or  machine,  we  would  find  on 
suddenly  causing  the  electrolysis  to  cease,  and  then  connecting 
the  two  silver  electrodes  through  a  voltmeter  that  there  was 
no  back  e.m.f.  produced.  The  cell  would  be  simply  in  the 
electrically  neutral  state  in  which  it  was  at  the  beginning.  It 
is  most  important  to  remember  that  the  osmotic  theory  of 
e.m.f.  refers  to  cells,  each  electrode  of  which  is  non-polarisable 


DEPOLARISA  TION  207 

or  perfectly  reversible.  If,  on  the  other  hand,  we  electrolysed 
a  dilute  sulphuric  acid  solution  (by  means  of  an  externally 
applied  e.m.f.),  using  platinum  electrodes,  and  then  cut  off  the 
external  current  and  joined  up  the  electrodes  as  before  we 
would  find  quite  a  considerable  e.m.f.  The  cell  is  then  said 
to  be  polarised.  What  has  happened  is  that  the  hydrogen  and 
oxygen  gases  which  have  been  produced  by  the  electrolysis  have 
formed  a  hydrogen  and  oxygen  electrode  respectively.  The 
existence  and  magnitude  of  this  back  e.m.f.  may  be  at  once 
observed,  if  the  externally  applied  e.m.f.  be  too  small,  for  if 
it  be  less  than  the  polarisation  e.m.f.  the  process  of  electrolysis 
would  automatically  stop.  A  platinum  electrode  dipping  into 
sulphuric  acid  is  an  example  of  a  non-reversible  or  polarisable 
electrode.  Of  course  once  the  platinum  has  become  charged 
up  with  either  hydrogen  or  oxygen  it  will  again  function  as 
a  reversible  hydrogen  or  oxygen  electrode,  the  platinum, 
however,  not  entering  into  the  phenomenon  as  such.  In  the 
above  case,  however,  there  is  polarisation  of  the  cell  as  a  whole, 
for  one  electrode  is  an  oxygen  electrode,  the  other  a  hydrogen 
electrode. 

A  polarisable  electrode  can  in  many  cases  be  converted 
into  a  non-polarisable  electrode  by  means  of  the  addition 
of  a  depolarizer.  Thus  mercury  in  contact  with  potassium 
chloride  solution  is  a  polarisable  electrode.  If,  however,  we 
saturate  the  solution  with  mercurous  chloride  (allowing  as  a 
guarantee  of  saturation,  a  layer  of  calomel  to  be  on  the  surface 
of  the  mercury),  the  electrode  mercury  |  mercurous  chloride 
-f-  potassium  chloride  is  now  non-polarisable.  This  particular 
electrode,  as  already  described,  is  known  as  the  calomel 
electrode.  Suppose  that  positive  current  is  passing  from  the 
solution  into  the  mercury.  This  means  that  electrons  are 
passing  from  the  mercury  into  the  solution.  An  electron 
leaves  the  mercury  and  attacks  a  mercury  ion  (present  from 
the  calomel),  thereby  discharging  it  so  that  it  is  deposited  as 
metallic  mercury.  This  means  that  at  the  same  moment  a 
chlorine  ion  discharges  itself,  at  the  second  electrode  or 
causes  some  secondary  reaction  to  take  place  at  this  second 
electrode,  which  causes  the  discharge  of  an  electron  on  to  the 


208       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

latter  electrode.  In  this  way  current  passes  through  the  cell, 
but  there  are  never  at  any  moment  any  unbalanced  ions 
present  in  the  solution  or  solutions  composing  the  cell.  As 
regards  the  mechanism  of  electrolysis,  attention  should  be 
drawn  to  some  rather  unexpected  results  obtained  by  F.  Haber 
and  J.  Zawadzki  (Zeit.  filr  physik.  Chem.,  78,  22  8,  1911)  in  the 
case  of  solid  compressed  salts  as  "  electrolytes." 


Electrometric  Method  of  Measuring  the  Solubility  of  Sparingly 
Soluble  Salts. 

In  Part  I.  (Vol.  I.),  the  electrical  method  of  determining  the 
solubility  of  a  salt  such  as  silver  chroinate  by  measurement  of 
the  conductivity  has  already  been  described.  In  the  case  of 
a  salt  like  silver  chloride  the  solubility  is  so  very  small  that 
the  conductivity  method  would  be  a  somewhat  inaccurate 
one.  By  means  of  electromotive  force  measurements,  how- 
ever, the  solubility  of  this  salt  can  be  determined  with  a  high 
degree  of  accuracy.  Thus  if  the  following  concentration  cell 
be  set  up — 


Ag 
electrode 


AgNO3     i    Saturated 
N  NH4NO3 


I00        i  with  silver  chloride 


N 
—  KC1  saturated 


10 


Ag 


the  current  is  found  to  flow  from  right  to  left  inside.  The 
silver  chloride,  which  we  can  regard  as  completely  dissociated, 
gives  rise  to  Ag'  and  Cl'  ions.  If  L  is  the  solubility  product 
[Ag"][Cr],  then  the  solubility  S  is  -\/L.  Now  in  the  presence 

N 

of  • —  KC1  (which  is  added  in  order  to  make  the  solution 
10 

conduct),  the  concentration  of  the  Cl'  ions  is  approximately 
0*1  N  (correctly  speaking  we  should  allow  for  the  fact  that  the 
degree  of  dissociation  is  not  complete),  and  practically  all  the 
Cl'  present  comes  from  the  KC1,  for  the  actual  quantity  of  Cl' 
produced  by  the  AgCl  is  quite  negligible  in  comparison. 
Hence  the  solubility  S  is  equal  to  -\/[Ag?]  X  o-i.  If  we  could 


SOLUBILITY  DETERMINATIONS  209 

determine  the  [Ag*]  the  value  of  S  could  be  at  once  calculated. 
If  the  above  cell  gives  an  e.m.f.  of  E,  it  is  clear  that — 

RT          o-o  i 
E  =  -—  loge  - 
n¥  x 

where  x  is  the  number  of  gram-equivalents  per  liter  of  Ag"  ions 

T?  T 

in  the  KC1  +  AgCl  solution.     The  factor  — ^  multiplied  by 

the  factor  converting  natural  logarithms  into  logarithms  to  the 
base  10,  has  the  value  0-058,  at  ordinary  temperatures,  if  the 
e.m.f.  is  to  be  given  in  volts.  That  is — 

E  volts  =  0*058  Iog10  — 

oc 

from  which  x  is  easily  calculated,  and  hence  the  solubility 
product  L,  and  from  that  the  solubility  S.  In  this  way  it  was 
found  that  the  solubility  of  AgCl  in  water  at  25°  C.  amounts 
to  1-2  X  io~5  gram-equivalents  per  liter.  (See  Goodwin, 
'/eitsch.physik.  Chem.,  13,  641,  1894.)  The  solubility  of  AgCl 
might  also  be  determined  by  measuring  the  e.m.f.  of  the  cell — 


I  N 

Ag  KC1  —  saturated  with  AgCl 


10 


Cl 


The  chlorine  electrode  consists  of  platinum  foil  saturated 
with  chlorine  gas.  If  both  electrodes  produced  cations,  the 
nett  e.m.f.  would  be  the  difference  of  the  two  single  potential 
differences.  Since,  however,  the  chlorine  electrode  produces 
anions,  the  two  electrodes  assist  one  another,  and  the  nett 
e.m.f.  of  the  cell  is  the  sum  of  the  two  single  potentials. 
That  is— 

E  =  0-058  log  ^Ag'  +  0-058  log  ^ 

'-Ag  L/C1 

and  since  —  0*058  log  CAB  and  —  0-058  log  Cci  represent  the 
electrolytic  potentials  of  these  two  electrodes,  which  we  can 
denote  by  TTAS  and  TTCI  respectively,  we  can  write — 

E  =  TTAg  +  7TC1  +  0-058  log 
T.P.C.— II. 


2io       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

(  N\ 

Knowing  E,  TTAg,  TTQI  and   ^ci'(  which  is  — J,  we  obtain  ^Ag-, 

and  as  before  the  solubility  product  of  dissociated  silver 
chloride.  If  the  above  cell  is  allowed  to  give  current,  the 
reaction  inside  the  cell  is  the  formation  of  silver  chloride, 
first  in  the  form  of  ions,  which,  of  course,  unite  to  give  solid 
silver  chloride,  since  the  solution  is  already  saturated  with 
respect  to  this  salt.  The  salt  is  formed  at  the  expense  of  the 
silver  and  chlorine  electrodes  respectively. 

Electrometric  Determination  of  the  Valency  of  Ions. 

The  classic  illustration  of  how  the  valency  of  an  ion  can 
be  determined  from  e.m.f.  measurements  is  that  of  the 
mercurous  ions  [Hg2++]  investigated  by  Ogg  (Zeitsch.  physik. 
Chcm.,  27,  285,  1898). 

The  mercurous  ion  may  be  represented  by  either  Hg+  or 
[Hg2++].  The  method  of  investigating  whether  the  mercurous 
ion  was  a  single  atom  carrying  one  charge  or  two  atoms 
together  carrying  two  charges,  is  determined  by  calculating  ?i 
in  the  e.m.f.  expression — 

RT.      r2 
10g 


Suppose  the  following  cell  is  set  up — 


Hg 


N 

—  Mercurous  nitrate 


2 


dissolved  in 

—  nitric  acid 
10 


N 

—  Mercurous  nitrate 


20 


dissolved  in 


N 


10 


nitaic  acid 


Hg 


+  P°l£ 


Inside. 


The   nitric   acid    is   present    to   prevent    hydrolysis   of    the 
mercury  salt.     Suppose  the  concentration  of  mercury  ions  in 

N 
-  mercurous  nitrate   is  represented   by  c\  and   that  in  the 


DETERMINATION  OF   VALENCY  211 

N 

-  by  <r2.     Neglecting  the  liquid  |  liquid  P.D.,  we  can  write 

the  e.m.f.  E  of  the  cell  in  the  form  — 
RT  ,       <:<> 


For  the  actual  case  mentioned  E  was  observed  to  be  0*029 
volts.     Taking   as   a   first    approximation    that-2  =  —  =  10, 

then  Iog10  —  =  unity,  and  — 
c\ 


o'029  =         -  ,    or     n  =  2 
n 

That  is  the  valency  of  the  mercurotis  ions  is  two.  That  is  the 
mercurous  ion  carries  two  charges,  and  we  must  therefore 
represent  it  by  Hg2++  since  we  know  that  one  equivalent  of 
mercury  is  united  to  one  equivalent  of  NO3  in  mercurous 
nitrate.  The  proper  formula  for  a  molecule  of  mercurous 
nitrate  is  therefore  Hg2(NO3)2. 

Electrometric  Determination  of  the  Hydrolysis  of  Salts -1 

In  dealing  with  various  methods  of  measurement  of  the 
hydrolysis  of  salts  by  water  in  Part  I.  (Vol.  I.),  reference  was 
made  to  a  method  depending  upon  e.m.f.  determinations. 
This  method  will  now  be  described.  It  consists  essentially  in 
using  a  hydrogen  electrode  in  the  solution  of  the  salt,  which 
is  partly  hydrolysed,  the  other  half  of  the  cell  being  a  calomel 
element,  the  two  halves  of  the  cell  being  connected  by  means 
of  a  saturated  solution  of  ammonium  nitrate  which  was  taken 
as  annulling  the  liquid  |  liquid  potential  difference.  From  the 
e.m.f.  value  obtained,  the  concentration  of  H'  ions  present  in 
the  salt  solution  is  calculated,  and  hence  the  degree  of 
hydrolysis.  This  method  is  particularly  suitable  when  the 
concentration  of  H*  ions  is  very  small,  in  fact  in  cases  in 
which  other  methods  would  be  inapplicable.  Its  applicability 

1  Cf.  H.  G.  Denham,  Jvurn.  Chem.  Soc.,  93,  41,  1908. 


212        A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

is,  however,  limited  b^  the  fact  that  it  cannot  be  employed  to 
determine  the  hydrolysis  of  salts,  the  metal  of  which  is  more 
noble  than  the  hydrogen  itself,  i.e.  it  cannot  be  used  in  the 
case  of  those  metals  like  copper,  silver,  mercury,  gold,  which 
would  be  precipitated  upon  the  platinum  electrode,  an  equiva- 
lent quantity  of  hydrogen  passing  into  the  ionic  state,  but  it 
can  be  used  for  salts  of  metals  such  as  aluminium,  nickel, 
cobalt,  zinc,  cadmium  magnesium,  barium  ;  also  one  cannot 
employ  the  hydrogen  electrode  in  the  case  of  cations  such  as 
ferric* ++  ions,  which  are  partly  reduced  by  the  hydrogen,  or 
in  the  case  of  reducible  anions  such  as  NO3',  C1O3'.  It  has 
been  used  with  success  by  Denham  (/.^.)  in  determining  the 
hydrolysis  of  such  salts  as  aluminium  chloride,  aluminium 
sulphate,  nickel  chloride,  nickel  sulphate,  cobalt  sulphate,  and 
aniline  hydrochloride  amongst  others.  These  salts  in  which 
the  base  is  weaker  than  the  acid,  are  hydrolysed  by  water, 
giving  rise  to  some  of  the  free  base  undissociated  and  some 
free  acid,  which  is  largely  dissociated  so  that  the  solutions 
react  acid.  Such  salts  of  polyvalent  metals  can  exhibit  the 
phenomenon  of  progressive  hydrolysis  into  several  stages,  as 
already  pointed  out  in  Part  I.  Each  one  of  these  stages  has 
its  own  characteristic  hydrolysis  constant.  For  a  discussion 
of  the  relation  of  these  to  one  another  Denham's  paper  must 
be  consulted.  As  an  illustration  of  the  method  of  applying 
the  hydrogen  electrode  to  the  measurement  of  hydrolysis,  we 
shall  take  the  case  of  aniline  hydrochloride,  which  is  hydrolysed 
according  to  the  equation — 

C6H5NH2HC1  +  H,O  =  C6H5NH3OH  +  HC1 

Anilinium  hydroxide  completely 

mainly  wwdissociated        dissociated  in 
dilute  solution 

If  i  gram-mole  of  aniline  hydrochloride  is  dissolved  in 

3C 

v  litres  of  water  and  a  fraction  x  is  hydrolysed,  then  -  repre- 
sents the  concentration  of  anilinium  hydroxide  and  likewise  of 
the  acid  produced,  which  is  identical  with  the  concentration 
of  H'  and  Cl'  since  the  acid  is  practically  completely  dis- 
sociated. Regarding  the  water  concentration  as  constant  and 


DETERMINATION  OF  HYDROLYSIS  213 

therefore  really  taken  account  of  in  the  hydrolysis  constant 
K,  we  have  for  the  above  reaction  when  equilibrium  is 
reached— 

<r2 

K=  __  — 


as  already  explained  in  Part  I.     Knowing  v  we  have  only  to 


/v» 


obtain  -  ,  which  is  numerically  identical  with  the  concentration 

of  H'  ions,  to  be  able  to  calculate  K.    The  single  potential  TT 
at  the  hydrogen  electrode  can  be  written  in  the  form  — 

RT         ctf 


or,  since  —  RT  log  C  is  simply  the  electrolytic  potential  (E.P.) 
which  we  can  denote  by  TTO,  and  is  also  known  from  other 
measurements,  we  can  write  — 

•   RT  i  .   RT  i      x 

w  =  wo  +  -S|rlog<H-  =  ir0  +  -sf  log- 

In  a  particular  case  when  the  aniline  hydrochloride  was 
made  up  to  v  =  32  litres  (at  25°  C.)  the  observed  e.m.f.  of 
the  cell— 

H2    —  C6H5NH2.HC1     Saturated          Normal        Mercury 


—  C6H5NH2.HC1 
3^ 

Saturated 
ammonium 
nitrate 

Normal 
KC1  saturated 
with  Hg2Cl2 

-^ 

was  0^4655  volt;  the  current  inside  the  cell  passing  from  left 
to  right,  so  that  the  normal  calomel  was  the  positive  pole  of 
the  cell.  Now  it  is  known  that  the  "  normal  calomel " 
possesses  a  potential  difference  between  the  mercury  and 
calomel-potassium  chloride  solution  of  -|-  0*56  volt1  approxi- 
mately, the  mercury  being  positively  charged  with  respect  to 
the  solution ;  hence  it  follows  that  the  P.D.  of  the  hydrogen 
electrode  must  be  0*56  —  0*4655  =  +  0*0946  volt,  the 
1  On  the  so-called  absolute  scale,  cf.  p.  203. 


2i4        A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

hydrogen  electrode  likewise  being  positive  with  respect  to 
the  solution  in  conta<?t  with  it.  Hence  — 

.   RTi      * 

0-0945  =  ^o+^p  log  - 

The  "  absolute  "  value  of  the  electrolytic  potential  of  hydrogen, 
viz.  TTO,  as  measured  against  the  "  normal  "  calomel  electrode 
(when  this  electrode  is  taken  not  as  zero  but  as  +0*5  6)  is 
+  0*277  volt,  that  is,  if  we  join  a  hydrogen  electrode  in 
contact  with  a  solution  of  normal  concentration  of  hydrogen 
ions,  with  a  normal  calomel  electrode  (ammonium  nitrate 
saturated  solution  being  interposed)  the  calomel  is  the  positive 
pole  of  the  cell,  the  current  flowing  inside  from  hydrogen  to 
calomel,  and  the  total  e.m.f.  of  the  cell  is  +0-283  volt-  [Note, 
if  we  take  the  calomel  as  zero,  it  is  obviously  0-283  volts 
higher  than  the  normal  hydrogen,  and  therefore  the  normal 
hydrogen  on  this  scale  would  be  represented  by  —  0*283  volt, 
as  has  been  done  in  the  table  of  Electrolytic  Potentials  (E.P.) 
values  already  given.]  It  follows  therefore,  since  n  —  i  for 

RT 

hydrogen  ions,  and  -=-  for  25°  C  is  0*059  [the  unit  of  energy 

being  the  volt-faraday,  and  the  natural  logarithm  being  trans- 
formed to  logarithm  to  the  base  10],  that  — 

°'°59  logic  f  =  °'°94-5  —  o'277  =  —  °'l825 
Hence  -  =  concentration  of  H'  ions  =  o-  00080  7  <°^.         j. 


Since  complete  hydrolysis  of  the  aniline  hydrochloride  would 
produce  a  value  for  -  of  ^  (assuming  complete  dissociation 

of  the  hydrochloric  acid),  the  percentage   hydrolysis  of  the 
aniline  hydrochloride  at  dilution  v  =  32  is— 

0-000807 

—  Y;  —  -  X  ioo  =  2-58 

32 

Further,  the  hydrolytic  constant  K  is  given  by  — 

(0-000807)2 

K  =  -  —  *  —          f^r  —  =  0-00002  1 
(i  —  0*000807)32 


DETERMINATION  OF  HYDROLYSIS 


215 


In  the  following  table  Denham's  figures  are  given  for  the 
percentage  hydrolysis  of  aniline  hydrochloride,  and  the  hydro- 
lytic  constant  for  a  series  of  dilutions — 


Dilution  of 
salt. 

V. 

Observed  e.m.f. 
of  cell. 

it 
Single  P.D.  of 
hydrogen  electrode. 

% 
Hydrolysis. 

Hydrolytic 
constant 
K  X  10  . 

16 

24 
32 

0-4567 
0*4609 

0'4655 

+  0-1033 
O-OQQI 
0-0945 

1-82 
2-32 
2-58 

2T 

2-3 

2*1 

Mean  =  2*16 

The  value  of  the  percentage  hydrolysis  for  v  =  32  is  2-58, 
which  agrees  well  with  that  found  by  Bredig  by  the  electrical 
conductivity  method,  namely,  2*61. 


CHAPTER    VIII 

Chemical  equilibrium  in  homogeneous  systems — Concentrated  solutions. 

EXPERIMENTAL   RESULTS  OBTAINED   IN  THE   MEASUREMENT 
OF  HIGH  OSMOTIC  PRESSURES. 

IN  Part  I.  (Vol.  I.)  it  was  pointed  out  that  in  the  case  of  dilute 
solutions ',  the  experimental  work  of  Morse  and  Frazer  and  their 
collaborators  had  shown  that  the  osmotic  pressure  actually 
measured  is  identical  within  the  limits  of  experimental  error, 
with  the  gas  pressure  as  predicted  by  the  van  't  Hoff  formula. 
That  is,  at  'constant  temperature  T  the  osmotic  pressure  is 
strictly  a  linear  function  of  the  concentration,  i.e.  P  =  RTC. 
When  the  solutions  begin  to  be  moderately  concentrated, 
however,  this  linear  relation  is  gradually  departed  from. 
Measurements  at  high  concentration  are  extremely  difficult  to 
make,  owing  to  the  difficulty  of  obtaining  membranes  suffi- 
ciently stout,  even  when  deposited  upon  porcelain  tubes,  to 
stand  the  great  difference  of  pressure.  Within  recent  years, 
however,  this  has  been  accomplished  in  the  classic  work  of  the 
Earl  of  Berkeley  arid  Mr.  E.  G.  J.  Hartley  (Phil.  Trans.,  206  A, 
481,  1906).  Their  method  of  measuring  P  was  to  determine 
the  "  equilibrium  pressure,"  that  is  the  pressure  which  must  be 
applied  to  the  solution  to  bring  about  a  state  of  equilibrium 
between  it  and  the  solvent,  so  that  no  solvent,  i.e.  water, 
passes  in  either  direction  as  a  whole  when  the  solution  and 
solvent  are  separated  by  a  semi-permeable  membrane.  In 
these  measurements  the  solvent  was  under  the  pressure  of  the 
atmosphere.  The  pressure  applied  to  the  solution  so  that 
there  is  no  movement  of  the  solvent  is  the  sum  of  the  "  equi- 
librium pressure,"  i.e.  P±  plus  the  pressure  exerted  on  the 
solvent,  'i.e.  the  atmosphere.  The  details  of  the  apparatus  and 


CONCENTRATED  SOLUTIONS 


217 


method  of  working  will  be  found  in  the  paper  referred  to. 
The  following  are  a  few  of  the  results  obtained  with  pure  cane 
sugar.  The  temperature  of  measurement  is  o°  C.  The  con- 
centration is  expressed  as  weight  of  sugar  in  grams  in  i  liter  of 
solution  prepared  at  15°  C.  The  solvent  is  at  i  atmosphere 
pressure. 

OSMOTIC  PRESSURE  MEASURED  AT  o°  C. 


1 80*  I  grams  per  liter 

300-2 

420-3 

540-4 
660-5 
750-6 


150 


P  ==    13*95  atmospheres 

P  =    26-77 

P  =    43-97 

P=    67-51 

P=  100-78 

P=  133-74 


300 


Concentrations  in,  grams 
per  litre   of  solution-. 

FIG.  64. 

These  results  give  some  idea  of  the  enormous  values 
actually  reached  in  concentrated  solutions.  The  curve  (Fig. 
64)  shows  the  osmotic  pressure  plotted  against 'concentration, 
so  as  to  show  the  departure  of  the  P  values  from  the  simple 
linear  relation,  which  holds  in  dilute  solutions.  The  straight 
line  is  drawn  on  the  usual  assumption  that  i  gram  molecular 
weight  of  solute  per  liter  (solution)  should  give  an  osmotic 
pressure  of  22*4  atmospheres. 

Other  sugars  were  also  employed,  and  in  later  papers  an 
account  is  given  of  similar  very  accurate  measurements  both 


218        A    SYSTEM   OF  PHYSICAL    CHEMISTRY 

of  osmotic  pressing  and  lowering  of  vapour  pressure,  due  to 
calcium  ferrocyanide  in  water,  this  salt  being  a  very  soluble 
one,  and  one  which  at  the  same  time  is  practically  stopped  by 
the  copper-ferrocyanide  semi-permeable  membrane  (Earl  of 
Berkeley,  E.  G.  J.  Hartley,  and  C.  V.  Burton,  Phil.  Trans., 
209  A,  177,  1909.  Dilute  solutions  of  the  same  solute  were 
also  investigated  by  the  Earl  of  Berkeley,  E.  G.  J.  Hartley,  and 
J.  Stephenson,  ibid.,  p.  319).  For  details  the  reader  is  again 
referred  to  the  original  papers.  The  object  of  the  work 
referred  to  was  to  test  Porter's  equation.  This  equation  wil 
be  taken  later. 


THEORETICAL  TREATMENT  OF  THE  OSMOTIC  PRESSURE 
OF  CONCENTRATED  SOLUTIONS. 

Sacku^s  Equation  of  State  for  Solutions  of  any 
Concentration. 


In  view  of  the  fact  that  the  simple  expression  PV  =  R 
breaks  down  for  concentrated  solutions  it  is  natural  to  expect 
that  attempts  would  be  made  to  account  for  the  behaviour,  by 
applications  of  gas  equations  such  as  that  of  van  der  Waals, 

viz.  (/  -f"  —  \v  —  b)=.  RT.     This  has  been  attempted  by  a 

V  ^2' 

fairly  large  number  of  workers,  with  only  partial  success,  how- 
ever. To  get  an  idea  of  this  method  of  treatment  the  reader 
should  consult  the  paper  of  O.  Stern  (Zeitsch.  physik.  Chew., 
81,  441,  19 1 2).1  Apart  from  the  fact  that  the  van  der  Waals 
equation  is  not  a  very  accurate  one  for  compressed,  i.e.  concen- 
trated, gases  and  therefore  may  be  expected  to  be  similarly 
inexact  for  solutions,  it  will  be  at  once  evident  that  4the 
"  constants  "  a  and  b  in  the  case  of  solutions  are  much  more 
complex  quantities  than  in  the  case  of  a  single  gas,  owing  t 
the  presence  of  different  kinds  of  molecules,  namely,  those  o: 
the  solvent  and  those  of  the  solute.  The  molecular  attractions 
which  are  taken  account  of  by  the  term  a  are  due — 

1  For  a  criticism  of  Stern's  paper  see  J.  J.  van  Laar  (Zeitsch,  physik. 
.y  82,  223,  1913). 


SACKUtfS  EQUATION  FOR  OSMOTIC  PRESSURE   219 

(1)  to  attractions  between  the  molecules  of  solvent  for  one 
another,  and 

(2)  attraction  between  the  molecules  of  solute   and   the 
molecules  of  solvent. 

Rather  remarkably,  however,  O.  Sackur  (Zeitsch.  physik. 
Chem.,  70,  477,  1910)  has  discovered  that  a  simplifitd  form 
of  van  der  Waals'  equation  actually  holds  good  with  con- 
siderable accuracy  up  to  very  high  concentrations.  The 
expression  is — 

P(V  -  b)  =  RT 
where  P  is  the  osmotic  pressure 

V  =  the  volume  of  solution  in  which  i  mole  of  solute  is 

dissolved 
R  =  gas  constant  1*985  calories  per  degree  or  0*082 

liter  atmospheres 
T  =  absolute  temperature. 

It  is  important  to  note  the  definition  of  V,  i.e.  volume  of 
solution,  for  in  concentrated  solution,  owing  to  changes  in 
volume  (contraction  or  expansion)  due  to  solute,  one  cannot 
assume  that  i  liter  of  solution  can  be  produced  by  adding 
the  required  mass  of  solute  to  i  liter  of  solvent.  In  the 
following  table  are  given  some  of  the  data  used  by  Sackur  to 
test  his  equation  in  the  domain  of  concentrated  solutions. 


Solvent :   Water.     Solute  :  Glucose  at  o°C.     (Measurements  of  Berkeley 
and  Hartley.) 


V. 

P  in  atmospheres. 

PV  (obs.). 

PV  calculated  by 
Sackur's  equation. 

1-805 

I3-2 

23-9 

24-4 

0-903 

29-2 

26-3 

26-8 

0-565 

53'2 

30-0 

30'4 

0-402 

87-8 

35  '2- 

35'5 

0-328 

I2I-2 

41*0 

40-6 

220       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 


Solvent :  Water.  Solute  :  Acetamide.  (Measurements  of  P  obtained  in- 
directly from  freezing  point  determinations  by  Jones  and  Getman, 
Amer.  Ckem.  Journ.,  32,  308,  1904.) 


P  atmospheres. 


23-5 

49-6 
79-6 

114 

I48'5 

203 

272 

347 

445 


PV  observed. 

PV  calculated  by  Sackur's 
equation. 

23-5 

23-7 

24-8 

25  '3 

26-5 

27-0 

28-5 

29-0 

29-7 

31-0 

38-9 

38-2 

43*4 

42'5 

49*4 

48-2 

Sackur  found  that  the  b  term  of  the  equation  showed,  in 
general,  an  increase,  as  solutes  of  higher  molecular  weight 
were  employed  in  the  same  solvent.  Further,  Sackur  found 
that  b  was  very  sensitive  to  temperature.  Thus  for  glucose 
at  o°  C.  &  =  o'i6,  at  22°  C.  ^  =  0*093.  He  considers  that 
this  is  due  to  the  hydration  of  solute  molecules,  the  degree  of 
hydration  varying  rapidly  with  temperature.  The  applicability 
of  the  simple  equation  P(V  —  b}  =  RT  shows  that  van  der 
Waals'  constant  «,  which  takes  account  of  molecular  attrac- 
tions, can  be  neglected  up  to  extremely  high  concentrations. 
Sackur  attempts  to  make  this  appear  at  least  plausible  on  the 
following  grounds.  The  apparent  lessening  of  the  pressure 
(directed  outwards)  due  to  the  attraction  of  the  molecules, 
inwards,  is  naturally  only  effective  at  the  surfaces  of  the  com- 
pressed gases,  or  pure  liquids  (as  these  attractive  forces  balance 
one  another  in  the  bulk  of  the  gas  or  liquid).  In  solutions, 
however,  there  exist,  besides  attraction  between  the  solute 
molecules  themselves,  attractions  also  between  the  solute 
molecules  and  the  solvent  molecules,  and  the  latter,  according 
to  Sackur,  "  are  probably  much  the  greater."  In  the  process 
of  performing  osmotic  work,  i.e.  in  the  process  of  diluting  a 
solution  by  the  addition  of  some  solvent,  some  work  is  done  in 
drawing  the  solute  molecules  apart  from  one  another,  i.e.  work 
against  the  first  type  of  attraction  ;  while,  on  the  other  hand, 


SACKUR'S    VAPOUR  PRESSURE   EQUATION      221 

the  process  of  dilution  is  actually  aided  by  the  attraction 
between  the  solvent  and  solute  molecules.  Since  these  two 
effects  are  mutually  opposed,  it  is  reasonable  to  expect  that 
the  total  attraction  effects  in  solution  will  only  become  notice- 
able at  much  higher  concentration  than  in  the  gaseous  state 
itself  (Bredig,  Zeitsch.  physik.  Chem.^  4,  44,  1889). 


CONNECTION  BETWEEN  THE  OSMOTIC  PRESSURE  OF  CONCEN- 
TRATED SOLUTIONS  AND  THE  RELATIVE  LOWERING  OF' 
VAPOUR  PRESSURE. 

This  problem  has  likewise  been  the  subject  of  much  theo- 
retical investigation,  generally  of  a  very  complicated  kind.  A 
full  discussion  will  not  be  attempted  here.  For  further  infor- 
mation the  reader  is  referred  to  the  treatment  of  the  subject 
and  papers  cited  by  Nernst  in  his  textbook  (English  edition, 
p.  155  seq.\  notably  those  of  McEwan  (Zeitsch. physik.  Chem.,  14, 
409,  1994),  Dieterici  (Wied.  Annalen,  48,  513,  1891;  50,  47, 
1893;  52,  263,  1894)  and  also  Noyes  (Zeitsch.  physik.  Chem., 
35,  707,  1900). 

In  the  present  instance  we  shall  restrict  ourselves  to  two 
lines  of  treatment,  namely  that  followed  by  Sackur  in  his  book 
Lehrbuch  der  Thermochemie  und  Thermodynamik^  p.  200  seq., 
and  also  the  more  general  comprehensive  theory  of  osmotic 
pressure  and  lowering  of  vapour  pressure  as  worked  out  by 
Prof.  A.  W.  Porter,  F.R.S.  (Part  I.,  Proc.  Roy.  Soc.t  79  A,  519, 
1907  ;  Part  II.,  Proc.  Roy.  Soc.,  80  A,  457,  1908). 

Sacknr's  modification  of  the  Eqiiation  connecting  Lowering  of 
Vapour  Pressure  and  the  Osmotic  Pressure  for  Solutions  of 
any  Concentration. 

In  dealing  with  the  lowering  of  vapour  pressure  of  dilute 
solutions,  the  following  formula  for  the  osmotic  pressure  P  was 
deduced. 


222        A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

The  vapour  was  supposed  to  behave  approximately  as  a 

j 

perfect  gas,  and  further  the  term  -~-  was  taken  to  be  the  density 

of  the  solvent  which  was  considered  to  be  sensibly  the  same 
as  that  of  the  solution,  i.e.  the  solution  was  so  dilute  that  on 
adding  a  volume  of  solvent  dv  the  volume  of  the  solution 
increased  exactly  by  this  amount.  If,  however,  the  solution  is 
concentrated,  addition  of  the  solvent  (volume  dv)  will  not 
cause  simply  this  increase  in  the  volume  of  the  solution.  The 
actual  increase  may  be  either  less  or  greater  than  dv.  The 

dx 

term  —  as  it  appears  in  the  thermodynamic  cycle  followed, 

must  be  replaced  by  a  slightly  more  complicated  function  in 
the  case  of  concentrated  solutions. 

Let  us  denote  by  p  the  density  in  grams  per  cubic  centi- 
meter of  a  given  solution,  which  contains  i  mole  of  solute  and 
x  moles  of  solvent  (reckoned  as  simple  molecules),  in  V  liters 
of  solution.  If  M  is  the  molecular  weight  of  the  dissolved 
solute  and  M0  the  molecular  weight  of  the  solvent,  then  we 
can  write — 

M  +  #M0  _ 
Viooo     ~P 

and  hence  dx  =  ~^jrt&N  +  W/>) 

dx iooo/ 

or  ITT  — :  ~n 


If  c  is  the  concentration  of  the  solute  in  moles  per  liter  (of 
solution)  then  c  =  ~,  and  hence — 

dx 
Hence   the   expression  for  the  osmotic  pressure  may  be 

H-pn — 


written — 


SACKUR'S    VAPOUR   PRESSURE  EQUATION     223 

For  a  dilute  solution  c  is  nearly  zero,  and  p  becomes  pQ 
the  density  of  the  solvent.     That  is — 


which  is  the  van  't  Hoff  equation  previously  deduced  with  a 
slight  alteration  in  the  concentration  units  included  in  the 

IOOO 

term  —     - .     Sackur  has  tested  the  above  relation  for  concen- 
M0 

trated  solutions ;  the  only  investigation  hitherto  carried  out 
which  afforded  sufficient  data  to  calculate  the  expression  com- 
pletely being  that  of  the  Earl  of  Berkeley  and  Hartley  (I.e.) 
on  solutions  of  calcium  ferrocyanide.  In  the  following  table 
taken  from  Sackur's  textbook,  is  given  the  comparison  of  the 
found  and  calculated  results. 


VAPOUR  PRESSURE  AND  OSMOTIC  PRESSURE  OF  CONCENTRATED 
CALCIUM  FERROCYANIDE  SOLUTIONS  AT  o°  C. 


Grams  of 

_  :n 

P  atmos. 

C'a2Fe(CN)c 
in  1000  grams 
of  water. 

p- 

moles  per 
liter. 

£ 

^  found. 
/ 

calculated 
by 
Sackur. 

P  atmos. 
found. 

3I3'9 

•224 

•oo 

0-195 

I'033 

40-7 

4I-22 

395go 

•270 

•23 

0-190 

1-057 

70  '8 

70-84 

428-9 

•287 

•32 

0-183 

1-070 

86-2 

87-09 

472-2 

•309 

'44 

0-181 

I-092 

114-0 

112-84 

499'7 

•322 

•51 

I-I07 

131-0 

I30-66 

As  Sackur  points  out,  this  relatively  simple  formula  gives 
very  satisfactory  agreement  between  observed  and  calculated 
values.  Berkeley  and  Hartley  have  deduced  a  more  com- 
plicated formula  which  will  be  referred  to  in  Porter's  theory 
(vide  infra\  in  which  compressibility  of  the  solution  was 
allowed  for  as  well.  In  the  above  formula  the  solution  is 
treated  as  though  incompressible,  the  change  in  density  with 
concentration  being  alone  allowed  for.  It  must  be  remembered 
also  that  the  vapour  (of  the  solvent)  has  been  treated  as  a 
perfect  gas,  and  the  solute  has  been  treated  as  absolutely 
non-volatile. 


224       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

Porters  Theory  of^  Compressible  Solutions  of  any  Degree  of 
Concentration. 

In  the  following  account,  Prof.  Porter's  own  words  are 
employed  practically  throughout.  It  has  been  considered 
sufficient  for  the  present  purpose  to  restrict  ourselves  to  the 
case  in  which  solvent  alone  is  volatile.  The  further  case, 
where  both  solute  and  solvent  are  volatile,  is  given  in  Porter's 
second  paper  (Lc.).  One  point  remains  to  be  noted,  namely, 
the  emphasis  which  is  laid  in  Porter's  theory  on  the  effect  of 
hydrostatic  pressure  upon  vapour  pressure.  It  is  a  well  known 
experimental  fact  that  if  we  add,  say,  an  inert  gas  to  a  vessel 
containing  a  liquid  and  its  saturated  vapour,  that  increase  in 
pressure  in  the  vessel  due  to  the  inert  gas,  causes  an  increase 
in  the  actual  pressure  (partial  pressure)  of  the  saturated 
vapour.  From  the  molecular  standpoint  one  might  regard  the 
phenomenon  as  due  to  the  closer  packing  of  the  molecules 
of  the  liquid  under  the  increased  pressure,  and  therefore  the 
possibility  exists  of  more  molecules  per  unit  surface  area 
capable  of  forming  vapour,  i.e.  the  vapour  pressure  rises. 
With  this  reference  by  the  way  we  can  now  go  on  with 
Porter's  paper. 

"This  paper  is  an  attempt  to  make  more  complete  the 
theory  of  solutions,  at  the  same  time  maintaining  as  great 
simplicity  of  treatment  as  is  possible  without  sacrificing  pre- 
cision. Renewed  attention  has  been  called  to  the  subject, 
owing  to  the  success  of  the  experiments  of  the  Earl  of  Berkeley 
and  Mr.  E.  J.  Hartley  on  the  osmotic  pressure  of  concentrated 
solutions  of  sugar.  Diversity  of  opinion  has  existed  in  regard 
to  the  interpretation  of  these  experiments,  insufficient  attention 
having  been  previously  paid  to  the  influence  of  the  hydro- 
static pressure  of  the  pure  solvent  upon  the  value  of  the 
osmotic  pressure.  The  principal  advances  made  in  this  paper 
consists  in  simply  demonstrating  the  influence  of  pressure  upon 
osmotic  pressure  for  compressible  solutions  and  in  including  the 
effect  of  the  variability  of  vapour  pressure  with  hydrostatic 
pressure.  The  influences  of  accidental  properties  (such  as  the 
effects  of  gravitation)  are  excluded. 


PORTERS  EQUATION  225 

"  Summary  of  Notation. 

"  The  following  is  the  notation  employed.     All  the  values 
are  isothermal  values. 
Solution  — 

Hydrostatic  pressure    .........    p 

Vapour  pressure  corresponding  to  hydrostatic 
pressure/   ............     TT> 

Vapour  pressure  when  solution  is  in  contact 
with  its  own  vapour  alone       ......     -nv 

Volume  at  hydrostatic  pressure  /    .....     Vp 

Reduction  of  volume  when  i  gram  of  solvent 
escapes  .............     sp 

Osmotic  pressure  for  hydrostatic  pressure/  .     .     Pp 

3)  »  J)  "V          •          •*•  7T 

Solvent  — 

Pressure  of  solvent  when  solution-is  at  pressure/  /0 

Corresponding  vapour  pressure       .....  7T0p 

Vapour  pressure  when  hydrostatic  pressure  is 

that  of  its  own  vapour    ........  TTOO 

Volume  at  hydrostatic  pressure  /0  .....  Vqp 

»  55  })          ^oo      ....  VQ^ 

Specific  volume  at  hydrostatic  pressure  /0    .     .  up<t 


5) 


Vapour  of  Solvent  — 
Specific  volume  at  pressure  TTOO  ......     z/0 


K      ......  TT 

A  few  special  symbols  are  defined  in  the  text. 

"  Relation  between  Osmotic  and   Vapour  Pressures. 

"  The  following  isothermal  cycle  enables  the  above  relation 
to  be  found.  A  large  (practically  infinite)  quantity  of  a  solu- 
tion (unacted  upon  by  any  bodily  field  of  force,  such  as  gravity) 
is  separated  from  a  quantity  of  pure  solvent  by  a  semi- 
permeable  membrane.  Tfa  solute  is  supposed  to  be  involatile. 

T.P.C.  —  II.  Q 


226       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

The  solution  is  under  a  hydrostatic  pressure  /,  while  the 
solvent  is  under  the  hydrostatic  pressure  pQ  for  which  there 
will  be  equilibrium.  It  is  not  intended  that  either  of  these 
pressures  shall  be  restricted  to  be  the  vapour  pressure  of  the 
corresponding  liquid.1 

"  (i)  Transfer  i  gram  of  solvent  from  the  solution  to  the 
solvent  by  moving  the  semi-permeable  membrane  to  the  left, 
in  Fig.  65  ;  the  work  done  upon  the  system  is  — 


"(2)  Separate  i  gram  of  the  pure  solvent  (at/0)  from  the 
rest  by  partitioning  off  the  lateral  tube  ;  change  its  pressure 


2  —  *  —  1 

J    ^ 

# 

<< 

p 

1 

Solution 

Solvent 

| 

<           Po 

i 

V 

V0 

j 

FIG.  65. 

to  TTOO  (by  aid  of  the  lateral  piston),  so  that  it  will  be  in 
equilibrium  with  its  own  vapour,  and  then  evaporate  it  j  the 
work  done  is — 


7TO 


pdu  —  7700(z/0  —  n 


11  (3)  Change  the  pressure  of  the  vapour  to  7rn>  so  that  it  may 
be  in  equilibrium  with  the  solution  when  under  the  hydro- 
static pressure  of  its  vapour  alone  ;  the  work  done  is  — 


—  |  fdv 

J   ITnn 


.  J  111  fact,  the  values  of  the  "natural"  vapour  pressures  go  in  the 
opposite  direction,  i.e.  the  vapour  pressure  TT-K  of  the  solution  is  less  than 
the  vapour  pressure  of  the  solvent  7T00,  whilst  /  is,  as  a  matter  of  fact, 
greater  than  p0 


PORTER'S  EQUATION  227 

"  (4)  Close  the  semi-permeable  membrane  which  separates 
the  solution  from  the  solvent  by  a  shutter  to  which  hydro- 
static pressure  /0  can  be  applied  ;  also  enclose  the  solution 
by  a  second  shutter,  to  which  a  pressure  p  may  be  applied 
[these  two  steps  do  not  involve  work]  ;  the  solution  may 
now  be  removed.  Change  its  pressure  to  TT^  bring  it  into 
contact  with  the  separated  vapour  of  the  solvent,  which  is 
also  at  a  pressure  TT^  ;  condense  this  vapour  into  it,  thereby 
increasing  the  volume  of  the  solution  by  ^v,  and  then  compress 
to  a  pressure/.  The  work  done  is  — 

-  rpdV  +  TT^  -  sj  +  f  X(V  +  s) 
J  p  J  p 

"The  connection  through  the  semi-permeable  membrane 
must  now  be  restored,  and  then  everything  will  be  in  its  initial 
state,  and  the  total  work  done,  since  the  cycle  is  isothermal, 
must  be  zero. 

"  Adding  the  several  terms,  integrating  by  parts,  and  simpli- 
fying this  equation,  we  obtain  — 


rPo  r*oo          (P 

-      *tdp-\   vdp  +  \    sty  = 

J  JT  J  V  J  ir 


or,  remembering  that  Bp=/  —  /0,  [since  equilibrium  is  main- 
tained, as  in  Fig.  65,  by  the  help  of/0  and/]— 

fP  /"TOO  rP-Vp 

I    sdp  =  \    vdp+\    udp    .     .     .     .     (i) 

J    TTjr  J    TTjr  J    7T00 

"  This  is  the  expression  which  gives  the  osmotic  pressure 
for  any  concentration  and  temperature  in  terms  of  the  vapour 
pressures,  etc.,  corresponding  to  the  same  concentration  and 
temperature.  It  includes  the  influence  of  compressibility,  and 
states  with  precision  the  particular  circumstances  to  which  the 
various  physical  data  correspond.  For  example,  the  vapour 
pressures  7r00  and  TT^  are  the  vapour  pressures  of  the  solvent 
and  solution  each  iinder  the  hydrostatic  pressure  of  its  own 
vapour,  and  not  under  the  hydrostatic  pressures  /0  a°d  p 
respectively,  as  might  perhaps  have  been  expected. 

"  In  order  to  compare  this  equation  with  those  hitherto 


228       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

given,  we  will  first  assume  that  s  and  u  are  constants  (that  is, 
we  ignore  compressibility),  and  that  the  vapour  follows  the  gas 
laws.  The  equation  then  becomes  — 


where  R  is  the  gas  constant  for  solvent  vapour.     This  may 
also  be  written  — 


(P*  +  A  -  **)*  +  (PP  +  *oo  -/)«  =  RT  log  (3) 

\  T/TT   / 

"  The  following  special  cases  are  of  interest. 

"  i  st.  Let  P,r0  be  the  osmotic  pressure  when  the  solvent  is 
under  the  hydrostatic  pressure  TTOO  of  its  own  vapour.  Then 
PTTO  =/  —  TT-QO,  so  that  (P»f  +  7700  —  p)u  =  o  ;  and 


(Pir.  +  "00-^  =  ^  log  .     .     (4) 

"  This  is  identical  with  van  't  Hoff's  case,  except  that  he 
writes  it  in  terms  of  molecular  quantities  and  pays  no  attention 
to  the  variation  of  P  and  TT  with  hydrostatic  pressure. 

"  2nd.  Let  P,r  be  the  osmotic  pressure  when  the  solution  is 
under  the  hydrostatic  pressure  TT^  of  its  own  vapour.     Then 

P,r  =  7T,,.  —  /0,  SO  that  (Pff  —  T 


(P.  +  7r00  -  7f,)u  =  RT  log  ^)     .     .     (5) 

"  This  is  identical  with  the  Earl  of  Berkeley's  solution,  in 
which,  however,  no  attention  was  paid  to  the  influence  of 
pressure.  It  is  precisely  the  result  naturally  given  by  the 
method  he  employs  when  attention  is  paid  to  pressure. 

"  Influence  of  Hydrostatic  Pressure  of  Solution  upon 
Osmotic  Pressure. 

"  By  differentiating  formula  (2)  with  respect  to  /,  the  con- 
centration (c)  and,  therefore,  the  value  of  n^  being  maintained 
constant,  we  get — 


-         or  = 

pdl~n'  \dp) 


PORTER'S   EQUATION 


229 


"  This  does  not  allow  for  compression. 
"By  differentiating  the  accurate  expression  equation  (i), 
we  get  — 


which  is  of  the  same  form  as  before,  but  the  terms  have  now 
more  precise  meanings.  Similarly,  the  rate  of  change  of 
osmotic  pressure  with  change  in  the  hydrostatic  pressure  of 
the  solvent  is  given  by 


(7) 


"  Comparison  of  Osmotic  Pressures  of  Solutions  of  different 
Substances  in  the  same  Solvent. 

[Comparing  two  solutions,  different  solutes,  same  solvent.] 
It  is  easy  to  show  that  the  two-fold  isotony  (for  vapour  and  for 
osmotic  pressures)  holds  for  any  hydrostatic  pressures  of  the 
solutions  (the  same  for  all),  provided  that  the  vapour  pressures 

be    measured    for    the    solutions  ^^_ _^^ 

when   imder  the  same  hydrostatic     ^^  ^^^ 

pressure.  This  can  be  shown  at 
once  by  considering  the  arrange- 
ment represented  in  Fig.  66. 

"Two  solutions  having  the 
same  solvent  are  contained  in  a 
vessel  and  separated  one  from 
the  other  by  a  semi-permeable 
membrane.  The  space  above 
contains  the  vapour  together  with 
an  inert  gas  whose  pressure  is 
A.  The  vessel  is  supposed  to  FlG-  66< 

be  in  a  region  free  from  gravitational  action.  Then  it  is 
obvious  that  if  the  osmotic  pressures  be  equal,  but  the  vapour 
pressures  be  different,  a  circulation  must  ensue  which  will 
upset  the  initial  osmotic  equilibrium  in  such  a  direction  as  to 
maintain  the  difference  of  vapour  pressures  and  thus  to  cause 
perpetual  flow ;  the  possibility  of  this  we  are  entitled  to  deny. 


A+flJ, 


Solution 

I 


Solution 
z 


230       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

"  In  order  to  show  how  this  result  is  consistent  with 
equation  (i),  it  is  necessary  to  find  the  mode  in  which  the 
vapour  pressures  vary  with  hydrostatic  pressure. 


"  Variation  of  Vapour  Pressure  with  Hydrostatic  Pressure. 

"  An   approximate   formula   for   this   variation    has    been 
obtained  by  Professor  J.  J.  Thomson  in  his  Applications  of 

Dynamics  to  Physics  and  Chemis- 
try^ by  means  of  the  Hamiltonian 
method. 

"  The  theorem  that  the  vapour 
pressure  of  the  pure  solvent  in- 
creases with  the  hydrostatic  pres- 
I    j  sure  can   be  obtained   in  a  very 

i"^  simple  way  as  follows  : — 

"  Let  a  vertical  tube  (Fig.  67) 
ZZfl  containing  the  solvent  be  enclosed 

in  a  closed  chamber  in  a  gravita- 
tional field  ,  and  let  equilibrium  be 
set  up.  Let  now  membranes  per- 
meable to  the  vapour  alone  be 
inserted  in  the  side  of  the  tube  at 
a  distance  apart  dh.  Let  /0  be 

the  hydrostatic  pressure  in  the  liquid  at  any  point,  and 
7T0  that  in  the  vapour.  Then  nPo  being  the  specific  volume 
of  the  liquid,  and  vn  that  of  the  vapour  at  the  corresponding 
pressures,  we  have — 


dh 


dh 


whence 


dire 


_-     UPO 

V-^ 


"  This  method  is  not  applicable  to  the  case  of  the  vapour 
of  a  solution,  because  the  concentration  of  the  solution  changes 
with  the  height. 

"  We  will  proceed  to  find  an  exact  formula  for  this  variation 


VAPOUR  AND   HYDROSTATIC  PRESSURES      231 


by  means  of  an  isothermal  thermodynamic  cycle,  consisting 
of  several  stages  : — 

"  A  large  volume  V  of  solution  is  taken  with  a  space  above 
containing  an  inert  gas  (say,  air)  and  vapour  enclosed  by  a 
piston  semi-permeable  to  the  vapour  alone,  which  is  again 
enclosed  by  a  non-permeable  piston  (Fig.  68).  The  semi- 
permeable  piston  will  experience  the  pressure  A  due  to  the 
inert  gas ;  the  pressure  on  the  non-permeable  piston  will  be 
the  pressure  of  the  vapour  alone,  which  is  TTP.  The  volume 
of  the  gas  and  vapour  is  initially  VA- 


-«  —  ^- 

: 

1 

V 

A+2T 

FIG.  68 

"(i)  Evaporate  i  gram  of  solvent  from  the  solution  by 
withdrawing  the  outer  piston,  leaving  the  inner  one  fixed; 
work  done  upon  the  system  in  this  process  is  equal  to — 

ASp  —  TTp(vnp  —  Sp)       Or        -  TTpV^  +pSp 

where  vv   is  the  specific  volume  of  the  vapour  at  the  pressure 
TTp,  and  p  =  A  +  TTP. 

"  (2)  Increase  the  total  pressure  to  /'  =  A7  -\-TTP>  by  moving 
both  pistons  such  amounts  that  no  further  liquid  condenses  or 
evaporates.  The  work  done  by  the  inner  piston  is — 


and  that  done  by  the  outer  piston  is — 


232        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

where  VB  is  a  volume  which  represents  the  fact  that  the  vapour 
which  at  the  first  pressure  was  to  the  left  of  the  inner  piston 
may  have  passed  through  it  on  the  change  of  pressure  taking 
place,  since  the  law  of  compressibility  of  the  vapour  will  not 
in  general  be  the  same  as  for  the  inert  gas  A. 

"(3)  Condense   i  gram  of  the  vapour   by  moving  outer 
piston  from  right  to  left,  keeping  inner  piston  fixed. 

Work  done  is  —  Tjyzv  ,  —  /'•*>' 

"(4)  Restore  the  original  state  of  the  system  by  suitably 
moving  the  two  pistons  ;  work  done  upon  the  system  is  — 

A/(N  +  VA)  +  f  *ird(V  +  VA  +  VB) 


where  JWVs  must  be  the  same  as  before. 

"  Since  the  above  represents  a  complete  isothermal  cycle 
the  total  work  is  zero  ;  that  is,  after  integrating  by  parts, 


(P  sdp  = 
J 


whence  —  -~  =  —     ......     (10) 


"  It  is  convenient  to  take  as  the  upper  limits  of  the  two 
integrals  — 

/'  =  7Tn       and        TTp'  =  77V 

"  This  result  is  for  a  solution  of  any  concentration  ;  hence, 
for  the  pure  solvent  we  have  — 


rPo 

Udp= 
J  P 


n 

dp     and  °  =        -.     •     (ii) 

Po  J  "oPo  WO  *<#• 

"This  last  result  is  identical  with  the  result  obtained  by 
Professor  J.  J.  Thomson  as  an  approximate  solution  ;  we  now 
see  that  it  is  accurate,  provided  that  precise  meanings  be  given 
to  the  variables  concerned. 


VAPOUR  AND  HYDROSTATIC  PRESSURES      233 

"  It  is  convenient  to  take  as  the  upper  limits  of  these 
integrals  /0'  =  TTOO  and  TTO  ,  =  TTOO. 

PO 

"By  means  of  these  equations  we  can  now  transform 
equation  (i). 

r«*  f"ir 

We  have  I   sdp  =  /    vdp 

J  p  J  *p 

Inserting  this  in  (i), 

,J>-Ep  fVp  ,*„  .np 

I    ndp  =  \    vdp+\    vdp=\    vdp    .     .     (12) 

*'  *oo  -'  "V  '  ""GO  '  ^oo 

"These  integrals  depend  only  upon  the  properties  of  the 
pure  solvent  and  upon  the  limits  of  integration. 

"  The  approximate  form  of  the  above  equation  is — 

(?,-/  +  iroo)«  =  RT  log 

"  When  /  is  the  value  for  which  the  hydrostatic  pressure  of 
the  solvent  is  7r00,  the  left-hand  side  of  this  is  zero ;  consequently, 
in  this  case  7r00  =  ir-p  (from  the  right-hand  side).  This  is 
simply  a  special  case  of  a  general  relation  to  be  proved  next. 

/P  fPo 

sdp  and   /    udp  into 
«ir  J  ^oo 

equation  (i) — 

fvP  /-"-00  rV°Po  f*P 

/    vdp  =  I    vdp  +  /    vap    or     I    vdp  =  o, 

'  TTT  J   TTT  ^^00  -.    *    Tflp0 

whence          TTp  =  TTO 

That  is,  when  a  solution  is  in  osmotic  equilibrium 
with  the  pure  solvent,  as  in  Fig.  65,  the  vapour  pres- 
sure of  the  solution  is  equal  to  the  vapour  pressure 
of  the  pure  solvent,  each  measured  for  the  actual 
hydrostatic  pressure  of  the  fluid  to  which  it  refers.1 

1  I.e.  the  solution  is  under  a  hydrostatic  pressure/,  the  solvent  is  under 
a  hydrostatic  pressure  /0,  such  hydrostatic  pressure  being  produced  by  an 
inert  gas. 

This  conclusion  is  one  of  the  most  important  points  of  Porter's  theory 
of  osmotic  pressure. 


234        A   SYSTEM  OF  PHYSICAL   CHEMISTRY 


"  That  this  is  so  is  almost  immediately  evident  from  the 

following  case  :  — 

"  The  solution  and  solvent  are  placed  in  a  vessel  and  sepa- 

rated by  a  semi-permeable  membrane  (Fig.  69).  The  space  above 

is  also  separated  into  two 
parts  by  a  partition  semi-per- 
meable to  the  vapour  of  the 
solvent,  but  not  to  an  inert 
gas.  A  pressure  difference 
p  —pQ  =  Pp  is  maintained 
between  the  two  sides  by  aid 
of  an  inert  gas.  Then,  unless 
the  vapour  pressures  TTP  and 


Solution 


Solvent 


TTO       are    equal,    a    flow    of 

Po 

vapour  will  occur  with  such 
consequent  evaporation  and 
condensation  on  the  two  fluids 

respectively  as  to  upset  the  initial  osmotic  equilibrium  in  a 
direction  which  will  maintain  the  difference  of  vapour  pressures 
and  thus  cause  perpetual  flow,  the  possibility  of  which  we  are 
entitled  to  deny.  This  conclusion  may  be  taken  as  a  check 
upon  the  equations  which  we  have  derived. 

"  We  have  considered  only  the  case  of  a  non-volatile  solute, 
but  it  is  easy  to  see  that  this  theorem  must  be  equally  true  if 
the  solute  is  volatile;  for  the  upper  partition  may  be  taken 
impermeable  to  the  vapour  of  the  solute  ;  and  the  argument  is, 
in  such  a  case,  in  no  way  changed. 

"  Standard  Conditions  of  Measurement. 

"  In  whatever  experimental  ways  osmotic  pressures  may  be 
determined,  it  is  necessary  to  decide  on  the  standard  conditions 
to  which  the  obtained  values  shall  be  reduced  for  the  purposes 
of  tabulation  and  comparison;  that  is,  to  what  hydrostatic 
pressure  shall  they  refer  ?  When  osmotic  pressures  are  com- 
pared by  De  Vries'  original  method,  as  they  still  often  are  (by 
means  of  vegetable  or  animal  cells),  the  solution  is  under  only 
a  moderate  pressure.  On  the  other  hand,  when  values  are 


LIQUID   MIXTURES  235 

obtained  by  the  method  adopted  by  the  Earl  of  Berkeley,  it  is 
\\\Qpure  solvent  that  is  under  a  moderate  pressure.  The  values 
of  the  osmotic  pressure  will  differ  in  general  in  the  two  cases. 

"  Now  it  seems  most  natural  to  reduce  always  either  to  the 
value  corresponding  to  the  solvent  under  its  own  vapour  alone 
or  to  that  corresponding  to  the  solution  under  its  own  vapour 
alone ;  and  of  these  two,  the  latter  seems  the  better.  It  is 
indeed  most  natural  of  all  to  think  of  the  osmotic  pressure  as 
being  a  property  of  the  solution  (just  as  its  vapour  pressure, 
volume,  etc.,  are),  the  pure  solvent  being  only  brought  into 
consideration  in  a  secondary  way  in  connection  with  an  experi- 
mental mode  of  determining  the  osmotic  pressure.  It  may  be 
objected  that  if  this  standard  be  adopted  the  equilibrium 
pressure  of  the  pure  solvent  will,  even  for  moderate  strengths 
of  solution,  usually  be  negative;  that  is,  the  solvent  would 
require  to  be  under  tension.  The  difficulty  is  relieved  when 
it  is  remembered  that  a  certain  amount  of  tension  in  liquids  is 
practically  possible,  and  the  osmotic  pressure  for  a  strong 
solution  might  always  be  conceived  as  being  measured  against 
a  less  strong  solution,  and  this  in  turn  against  a  less,  and  so 
on,  till  the  pure  solvent  was  reached.  If  this  standard  be 
adopted,  we  have,  from  equation  (i), 


[TOO  /•TOO 

I    udp  =  /    vdp 

"   *V-P7T        '  *» 


an  equation  which  is  capable  of  being  graphically  represented 
on  the  indicator  diagram  for  the  pure  solvent  (Fig.  70). 

"  The  equation,  in  fact,  states  that  the  hatched  area  must  be 
taken  equal  to  the  dotted  area;  the  vertical  height  of  the 
former  then  gives  the  osmotic  pressure." 

LIQUID  MIXTURES. 

These  are  solutions  in  which  the  vapour  pressure  of  the 
solute  cannot  be  regarded  as  negligible  compared  to  that  of 
the  solvent — in  fact  the  use  of  the  terms  solute  and  solvent 
are  not  very  applicable,  as  both  liquids  may  be  present  in  large 
amount.  Such  systems  have  been  examined  theoretically  and 


236       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

practically  by  Dolezalejt  (Zeitsch.  physik.  C/iem.,  64,  727,  1908; 
ibid.,  71,  198,  1910) ;  see  also  Stern,  loc.  cit.,  and  Washburn 
(Journ.  Amer.  Chem.  Soc.^  32,  670,  1910).  The  following 
simple  relation  has  been  put  forward  by  Dolezalek  as  being  of 
fundamental  importance  in  these  cases.  The  partial  vapour 
pressure  /a  of  a  component  a  present  in  the  liquid  state  is 
proportional  to  the  molecular  concentration  of  the  component 
a  in  the  liquid  state  when  such  concentration  is  expressed 
as  a  fraction  of  the  total  number  of  moles  present.  Tf  xn  is 


FIG.  70. 

the  fractional  molar  concentration  of  a  in  the  liquid  state 
at  a  given  temperature,  and  pQ  is  the  vapour  pressure  which  the 
liquid  a  alone  would  exert,  then  the  partial  pressure  of  a  in 
the  mixed  vapour  is  given  by  pa  where  — 


Washburn  (loc.  tit.  and  Trans.  Amer.  Electrochem.  Soc.t  22, 
330,  1912)  has  shown  in  the  following  manner  that  this  simple 
relation  can  only  hold  if  the  liquids  composing  the  system  are 


LIQUID   MIXTURES 


237 


perfectly  miscible.  Thus,  suppose  we  had  a  system  consisting 
of  two  liquid  layers  in  a  tube  (Fig.  71)  in  equilibrium,  the 
component  a  being  present  in  each  layer.  On  the  above 
assumption  we  will  have  for  the  vapour  pressure  of  a  above 
the  one  layer  the  expression  —  - 


and  similarly  for  the  other  the  vapour  pressure  of  a  is  — 

Au  =  A*n 

But  since  perpetual  motion  ((distillation  of  a  vi&  the 
vapour,  from  one  layer  to  the  other)  is  impossible,  it  is 
necessary  (by  the  Second  Law 
of  Thermodynamics)  that 

At  =  AM  and   nence  #»!=  xau. 

That  is,  the  molar  concentra- 

tion  of   a   in   both   layers   is 

identical.     But  this  is  applica- 

ble to  all  the  molecular  species 

present    in    each    layer,    and 

hence  both  layers  are  identical, 

and  cannot  therefore  form  two 

phases.      We    must    therefore 

conclude  that  the  above  simple 

vapour  pressure  law  only  holds 

for  mixtures  composed  of  perfectly  miscible  liquids  and  cannot 

apply  to  the  cases   in  which  partial  miscibility  exists.     The 

problem  of  liquid  mixtures  is  in  a  rather  rudimentary  stage 

at  present,  and  further  discussion  of  it  in  a  book  of  this  kind 

must  be  omitted.1 

1  In  addition  to  the  references  given,  the  reader  should  consult  the 
papers  of  Rosanoffand  Easley  (Journ.  Amer.  Chem.  Soc.,  31,  953>  I9°9  > 
Zeitsch.  physik.  diem.,  68,  641,  1910),  and  especially  the  chapter  devoted 
to  the  subject  in  Partington's  Thermodynamics. 


FIG.  71. 


CHAPTER   IX 

Equilibrium  in  heterogeneous  systems,  in  the  absence  of  electrical,  capillary 
or  gravitational  effects. — The  Phase  Rule  and  some  of  its  applications. 
— The  Theory  of  Allotropy. 

IN  Part  I.  (Vol.  I.)  the  question  of  heterogeneous  equilibrium, 
i.e.  equilibrium  in  a  system  consisting  of  more  than  one  phase, 
was  considered  from  the  kinetic  standpoint,  the  generalisation 
employed  being  the  Distribution  Law  of  Nernst.  We  now 
take  up  the  study  of  heterogeneous  equilibrium  from  the  stand- 
point of  thermodynamics. 


THERMODYNAMICAL  DEDUCTION  OF  THE  NERNST 
DISTRIBUTION  LAW. 

We  can  deduce  the  Distribution  Law  by  means  of  a  simple 
thermodynamic  cycle  carried  out  at  constant  temperature. 


Benzene 

Anil  me  Cj 


Water 
Aniline 


System 
II. 


FIG.  72. 

Thus,  consider  the  two  equilibrium  systems  I.  and  II. 
(Fig.  72)  each  consisting  of  two  phases  in  contact  (say,  water 
and  benzene)  with  a  solute  (say,  aniline)  distributed  between 
each  pair  of  phases.  Suppose  in  system  I.,  the  concentration 
of  the  aniline  in  the  water  and  benzene  is  ^  and  <r2  respectively. 


THE  DISTRIBUTION  LAW  239 

In  system  II.  the  concentrations  are  <r3  and  c±  respectively. 
The  systems  are  both  at  the  same  temperature.  Suppose  the 
molecular  state  of  the  aniline  is  the  same  in  the  water  as  it  is 
in  the  benzene,  and  further  suppose  that  the  solutions  are  all 
sufficiently  dilute  to  allow  of  the  application  of  the  gas  law. 
It  is  required  to  show  that — 

—  =  —  =  constant 

ist  Step. — Suppose  a  small  quantity  8n  gram-moles  of 
aniline  are  removed  from  the  water  solution  in  I.  and  trans- 
ferred to  the  water  solution  in  II.  isothermally  and  reversibly. 
The  maximum  work  done  at  constant  temperature  (and 
practically  constant  volume)  is — 

8«RT  log  ^ 

2nd  Step. — Now  suppose  the  quantity  8n  at  cz  passes  at 
constant  temperature  and  volume  into  the  benzene  layer  at 
concentration  c±.  No  work  is  done,  since  the  system  (II.)  is 
in  equilibrium. 

yd  Step. — Transfer  the  quantity  8n  from  the  benzene 
layer  in  II.  to  the  benzene  layer  in  I.  The  maximum  work 
done  is — 

8«RT  log  ^ 

tfh  Step. — Allow  the  quantity  8n  to  pass  into  the  water 
layer  in  I.  Since  the  system  I.  is  in  equilibrium,  this  transfer 
at  constant  temperature  and  volume  involves  no  work.  The 
cycle  is  now  completed,  and  since  it  has  been  carried  out 
isothermally  and  reversibly  it  follows  from  the  Second  Law 
that  the  total  work  is  zero.  That  is — 

8»RT  log  ^  +  8»RT  log  -^  =  o 
or  -l  =  -i     or      —  =  —  =  constant 

which  was  to  be  proved* 


240        A    SYSTEM   OF  PHYSICAL   CHEMISTRY 

Having  already  given  considerable  experimental  illustra- 
tions of  the  principle  oT  the  Distribution  Law,  it  is  unnecessary 
to  consider  it  further  here.  Instead  we  shall  take  up  the 
subject  of  heterogeneous  equilibrium  from  a  much  more 
general  standpoint. 


THE  PHASE  RULE. 

Following  this  method  we  arrive  at  an  important  gene- 
ralisation called  the  Phase  Law,  or  Phase  Rule,  first  deduced 
by  Willard  Gibbs  in  1878  (cf.  Gibbs'  Scientific  Papers, 
published  by  Messrs.  Longmans)  by  means  of  his  "  chemical 
potential  "  method.  Gibbs'  Rule  received,  however,  no  practical 
application  until  it  was  taken  up  by  the  Dutch  physical  chemist, 
Bakhuis  Roozeboom,  who  showed  in  a  series  of  classical 
researches,  the  fundamental  importance  of  the  principle  as  a 
guide  to  the  behaviour  of  heterogeneous  chemical  systems  (cf. 
B.  Roozeboom's  book,  Heterogene  Gleichgewichte,  which  has 
recently  been  re-edited  and  extended  by  Meyerhoffer).  The 
original  method  of  deducing  the  Phase  Rule  employed  by 
Gibbs  is  by  no  means  simple,  and  many  alternative  methods 
have  since  been  described  by  various  authors.  A  good  method 
is  given  in  Nernst's  Theoretical  Chemistry.  In  the  follow- 
ing pages  an  attempt  is  made  to  deduce  the  principle,  first  of 
all  by  a  very  simple  method  based  on  that  suggested  by  J.  A. 
Muller  (Comptes  Rendus^  146,  866, 1908),  and  secondly  a  more 
rigid  thermodynamical  method  is  given  for  those  who  have 
followed  the  chapter  on  more  advanced  thermodynamic  prin- 
ciples in  Chap.  II.  of  this  book.  It  must  be  emphasised  that 
the  Phase  Rule  per  se,  while  extraordinarily  useful,  could  not 
hold  its  important  position  were  it  not  that  at  the  same  time  we 
make  use  of  the  Le  Chatelier-Braun  principle  of  "  mobile 
equilibrium."  The  importance  of  the  simultaneous  application 
of  both  principles  has  been  emphasised  by  W.  D.  Bancroft,  in 
the  introduction  to  his  book,  The  Phase  Rule.  Within  the 
confines  of  a  single  chapter  it  is  obviously  impossible  to  give 
more  than  the  briefest  outline  of  the  applicability  of  the  Phase 
Rule  to  a  very  few  typical  examples.  In  order  to  grasp  the 


THE   PHASE   RULE  241 

full  significance  of  the  Phase  Rule,  the  reader  is  therefore 
recommended,  after  reading  this  chapter,  to  consult  the  work 
of  Findlay  on  The  Phase  Rule  and  Desch's  Metallography,  in 
this  series  of  textbooks,  and,  above  all,  Roozeboom's  book 
already  referred  to. 

Before  proceeding  to  consider  the  Phase  Rule  itself,  it  is 
necessary  to   obtain  a  clear  idea   of  what   is  meant  by   the 
terms  phase,  component,  and  degrees  of  freedom.     Findlay  (l.c.) 
defines  them  thus  : — "  A  heterogeneous  system  is  made  up  of 
different  portions,  each  in  itself  homogeneous  but  marked  off 
in  space,  and  separated  from  the  other  portions  by  boundary 
surfaces.   These  homogeneous,  physically  distinct  and  mechani- 
cally separable  portions  are  called  phases"     Thus  ice,  liquid 
water,   and   vapour   (steam)    are   three   phases   of  the   same 
chemical  substance,  water.      A  system  may  be  made  up   of 
any  number  of  coexisting  phases.   It  is  important  to  remember, 
however,  that  in  any  system  there  can  never  be  more  than  one 
vapour  phase,  because  all  vapours  and  gases  are  miscible  in  all 
proportions.      Thus   take   the   classic   case  of  heterogeneous 
equilibrium,   namely,  the  dissociation  of  calcium   carbonate. 
The  system  here  considered  is  made  up  of  two  solid  phases, 
i.e.  calcium  carbonate  and  lime,  one  gas  or  vapour  phase  which 
is  practically  entirely  carbon  dioxide,  though  we  must  imagine 
that  there  are  a  few  molecules  of  lime  and  calcium  carbonate 
present,  since   all  substances   have  a   vapour  pressure,  even 
though  it  is  extremely  small  in  the  case  of  most  solids.     This 
system  consists  therefore  of  three  phases.     Again,  consider 
the  case  of  ammonium  chloride  partly  in  the  solid  and  partly 
in  the  gaseous  state.     There  is  one  solid  phase,  namely  solid 
ammonium  chloride.     There  is  one  vapour  state  consisting  of 
ammonia  gas  and  hydrochloric  acid  gas,  and  a  small  quantity 
of  undissociated  ammonium   chloride   vapour.     This   hetero- 
geneous system  consists  of  two  phases.    The  system  containing 
a  solid  substance  in  contact  with  a  saturated  solution  of  that 
substance,    and   vapour   above   the   liquid,  consists    of  three 
phases,  one  solid,  one  liquid  (the  solution),  and  one  vapour. 
It  is  important  to  bear  in  mind  that  in  phase  equilibrium  the 
equilibrium  is  independent  of  the  absolute  mass  or  amount  of  the 

T.P.C.— II.  R 


242        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

phases.  Thus  in  the  case  of  calcium  carbonate  in  equilibrium 
with  lime  and  carbon  dioxide,  the  position  of  equilibrium 
(determined,  say,  by  the  vapour  pressure  value  as  we  saw  from 
the  standpoint  of  the  Law  of  Mass  Action)  is  quite  unaltered 
by  further  addition  of  calcium  carbonate  or  lime.  The  vapour 
pressure  of  a  liquid  is  unaltered  by  increasing  or  diminishing 
the  quantity  of  the  liquid.  Also  the  concentration  (saturated) 
of  solid  dissolved  by  a  given  liquid  is  independent  of  the 
quantity  of  solid  in  contact  with  the  liquid.  If  in  the 

CaCO3  —  CaO  —  CO2 

system  we  had  simply  increased  the  volume  of  the  gas  phase 
by  expanding  the  containing  vessel,  some  more  of  the  CaCO3 
would  dissociate,  the  pressure  would  again  take  on  its  original 
value  although  the  absolute  amount  of  the  vapour  phase  is 
greater  than  in  the  first  state.  Similarly,  if  we  diminish  the 
absolute  amount  of  the  vapour  phase  in  the  above  case  by 
diminishing  the  volume  of  the  vessel,  we  leave  the  equilibrium 
point  unchanged.  Also  if  CO2  gas  were  added  to  the  system, 
the  pressure  would  remain  constant,  i.e.  the  equilibrium  would 
be  unaltered,  as  long  as  any  CaO  remained  unchanged  into 
carbonate  (since  the  formation  of  solid  CaCO3  is  the  process 
which  occurs  in  order  to  keep  the  CO2  pressure  constant). 
When  enough  CO2  had  been  added  to  change  all  the  CaO  into 
CaCO3,  the  gas  pressure  would  begin  to  rise  (on  further 
addition  of  CO2),  but  this  alteration  in  the  equilibrium  point 
is  not  unexpected,  when  we  consider  that  we  have  altered 
the  nature  of  the  system  by  causing  one  of  the  phases  (CaO) 
to  entirely  disappear  as  such.  Under  these  circumstances  the 
alteration  in  pressure  on  adding  the  CO2  is  exactly  what  we 
would  predict  on  the  basis  of  the  Phase  Rule. 

Having  made  clear  what  is  meant  by  the  term  phase,  we 
have  to  consider  the  meaning  of  another  term — component.  A 
phase  is  made  up  of  one  or  more  chemical  substances  or  con- 
stituents. It  might  be  thought  that  these  constituents  also 
represent  the  components  of  the  system.  This,  however, 
is  only  partly  correct.  By  the  term  components  is  meant 
those  constituents  the  concentration  of  which  can  undergo 


THE  PHASE  RULE  243 

independent  variation  in  the  different  phases.  Findlay  (/.<:.) 
gives  the  following  definition  : — 

"  As  the  components  of  a  system  there  are  to  be  chosen  the 
smallest  number  of  independently  variable  constituents,  by  means 
of  which  the  composition  of  each  phase  participating  in  the  state 
of  equilibrium  can  be  expressed  in  the  form  of  a  chemical  equation" 

The  idea  will  be  made  clearer  by  considering  a  few 
examples. 

Take  the  case  of  the  system  liquid  water  and  steam  in 
equilibrium.  The  number  of  components  is  one,  namely  the 
chemical  constituent  H2O.  The  system  consists  of  two  phases, 
namely  liquid  and  vapour,  but  the  one  component  (H2O)  is  all 
that  is  necessary  to  cause  the  formation  of  either  phase.  At 
the  so-called  triple  point  at  which  we  have  ice,  liquid  water  and 
vapour  coexisting,  again  the  system  is  a  one-component  one, 
although  existing  in  three  phases.  The  actual  chemical  con- 
stituents, say,  in  the  solid  and  liquid  phases,  can  be  represented 
by  (H2O)3  and  (H2O)2  or  other  polymers,  but  these  are  all 
directly  produced  from  the  one  component  H2O.  Take 
another  case,  namely  the  system  consisting  of  copper  sul- 
phate penta-hydrate  and  trihydrate,  in  contact  equilibrium  with 
water  vapour.  The  solid  phases  are  CuSO4  .  5H2O  and 
CuSO4 .  sH2O.  The  vapour  phase  consists  practically  entirely 
of  H2O,  namely  water  vapour.  The  number  of  phases  is 
three,  one  vapour  and  two  solid.  The  number  of  compo- 
nents (or  independent  variable  constituents)  is  two,  namely, 
CuSO4  and  H2O.  Out  of  these  two  components  we  can  build 
up  each  phase.  Thus  the  penta-hydrate  solid  is  made  up  of 
one  molecule  of  CuSO4  and  five  molecules  of  H2O.  The  solid 
trihydrate  is  made  up  of  one  molecule  of  CuSO4  and  three 
molecules  of  H2O.  Note  that  the  ratio  of  the  molecules  of 
CuSO4  and  H2O  must  be  taken  as  independent.  It  would  not 
do  to  take  .them  in  a  fixed  ratio,  say  i  to  5,  for  this  would 
allow  us  to  form  the  penta-hydrate,  but  not  the  trihydrate. 
If  we  had  assumed  their  ratio  constant  we  would  be  dealing 
with  one  component  only,  namely  (CuSO4  .  5H2O).  This  is 
not  sufficient  for  the  present  case.  The  vapour  phase  can  also 
be  chemically  represented  by  the  two  components  named  by 


244       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

taking  zero  number  of  CuSO4  molecules  together  with  one 
H2O.  Again,  take  the  similar  case  of  the  CaCO3 — CaO — CO2 
equilibrium.  Here  again  we  have  a  /7#<?-component  system. 
The  components  may  be  either  CaO  and  CO2  or  CaCO3  and 
CO2.  Take  the  first  choice,  namely  CaO  and  CO2,  thus  the 
solid  CaCO3  phase  can  be  represented  by  one  molecule  of 
CaO  +  one  molecule  of  CO2.  The  CaO  solid  phase  can  be 
represented  by  one  molecule  of  CaO  and  zero  molecules  of 
CO2.  The  vapour  phase  can  be  represented  by  one  mole  of 
CO2  and  zero  molecules  of  CaO.  (Note  that  zero  as  well  as 
positive  and  negative  values  are  permissible.)  In  this  case 
also,  in  order  to  represent  each  phase,  we  must  again 
regard  the  CaO  and  CO2  as  independently  variable,  i.e.  the 
system  consists  of  two  components.  Now  take  the  case  of 
solid  ammonium  chloride,  in  contact  with  its  vapour  which 
consists  of  some  NH4C1  and  mainly  of  NH3  and  HC1  in 
equivalent  molecular  proportions.  The  number  of  components 
in  this  case  is  one,  namely  the  chemical  entity  ammonium 
chloride.  Thus  the  solid  phase  is  represented  by  NH4C1. 
The  vapour  phase  is  also  represented  by  the  formula  NH4C1, 
for  although  we  have  two  gases  present,  namely  NH3  and  HC1, 
these  are  in  the  stoichiometric  ratio  represented  by  the  formula 
NH4C1.  The  undissociated  molecules  of  ammonium  chloride 
present  in  the  vapour  are  simultaneously  represented  by 
NH4C1.  This  is  a  good  illustration  of  the  distinction  between 
chemical  constituents  and  components.  In  the  vapour,  for 
example,  we  have  three  chemical  constituents,  namely  am- 
monium chloride  undissociated  gas,  ammonia  gas,  and  hydro- 
chloric acid  gas,  but  they  are  all  formed  from  one  component, 
ammonium  chloride.  In  this  case  it  is  unnecessary  to  assume 
two  independent  constituents,  i.e.  components,  say  NH3  and 
HC1,  for  the  composition  of  both  phases  can  be  represented 
by  the  two  substances  NH3  and  HC1  in  a  fixed  molecular  ratio 

i  mo  e  3  They  neecj  not  be  independently  variable.  This 
i  mole  HC1 

will  illustrate  the  significance  of  the  term  "smallest  number" 
of  independently  variable  constituents  in  the  definition  of 
component.  A  further  important  point  may  be  illustrated  by 


THE  PHASE  RULE  245 

the  ammonium  chloride  case.  Suppose  we  add  NH3  or  HC1 
gas,  so  that  one  or  other  of  these  is  in  excess  in  the  gas  phase, 
then  the  composition  of  the  gas  phase  can  be  no  longer  repre- 

i  mole  NHo 

sented  by  the  ratio  -        — ^>,,,  *•<?.  by  the  one  component 
i  mole  HC1 

(NH4C1),  and  we  have  now  to  regard  the  system  as  containing 
two  components,  NH3  and  HC1.  These  being  independently 
variable  constituents,  we  can  form  all  the  phases  by  their 
means.  Under  these  conditions  the  ammonium  chloride  dis- 
sociation becomes  analogous  to  the  calcium  carbonate  or 
copper  sulphate-hydrate  equilibrium.  As  regards  nomenclature 
it  is  usual  to  denote  the  number  of  phases  of  a  system  by  r 
and  the  number  of  components  by  n.  We  may  conclude  this 
discussion  of  what  is  meant  by  the  components  of  a  system, 
by  quoting  another  definition  of  component,  namely  that  given 
by  Kuenen  (Proc.  Roy.  Soc.  Edin.^  23,  p.  317,  1899-1900). 

"  In  determining  n  we  must  not  count  separately  those 
substances  which  in  all  the  phases  (either  separately  or  in 
combination  with  others  in  the  ratio  in  which  they  occur  in 
the  same  phase)  may  be  formed  out  of  those  that  have  already 
been  counted,  with  the  additional  understanding  that  if  we 
obtain  different  results  for  the  total  number,  by  counting  in 
a  different  order,  we  are  to  take  the  smallest  of  the  numbers 
found." 

Having  defined  the  terms  phase  and  component  we  have 
now  to  consider  a  further  term,  namely,  the  degrees  of  freedom  1 
of  a  system. 

In  deducing  the  conditions  for  equilibrium  in  a  hetero- 
geneous system,  i.e.  in  deducing  the  Phase  Rule,  Gibbs 
regarded  every  system  as  defined  by  the  three  independent 
factors  or  variables — temperature,  pressure,  and  the  concentra- 
tions of  the  components  in  each  phase.  Now  in  dealing  with 
equilibrium  from  the  standpoint  of  thermodynamics,  there  are 
certain  "  thermodynamic  criteria  of  equilibrium,"  such  as  those 

1  The  term  degrees  of  freedom  in  the  present  case  must  not  be  con- 
fused with  the  same  term  already  referred  to  in  dealing  with  the  possible 
modes  of  motion  of  molecules  and  atoms.  Cf.  the  section  dealing  with 
specific  heat  and  energy  quanta,  Part  III.  Chap.  II. 


246       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

with  which  we  have  already  become  acquainted.  These 
criteria  take  the  form  of  thermodynamic  equations  containing 
terms  called  thermodynamic  potentials  characteristic  of  each 
component,  the  equilibrium  being  reached  when  these 
potentials  reach  certain  values.  (One  such  potential  of  which 
we  have  made  frequent  use  is  the  free  energy  of  a  component. 
This,  however,  cannot  be  conveniently  employed  here.)  By 
means  of  these  equations  it  is  possible  to  fix  the  values  of  a 
certain  number  of  the  variables  of  the  system,  in  fact  as  many 
variables  as  there  are  equations.  But  the  number  of  variables 
actually  possessed  by  a  system  may  be  greater  than  the 
number  of  thermodynamic  equations.  The  difference  between 
the  number  of  variables  and  the  number  of  equations  gives 
the  number  of  variables  which  are  left  undefined  by  the  equa- 
tions. These  variables,  which  are  really  variable  at  will,  are 
called  the  degrees  of  freedom  of  the  system.  Of  course,  as  long 
as  no  definite  values  are  assigned  to  these  variables  the  equili- 
brium of  the  system  as  a  whole  will  remain  undetermined. 
Findlay's  definition  (I.e.)  of  degrees  of  freedom  of  a  system  is  as 
follows :  "  The  number  of  degrees  of  freedom  of  a  system  are 
the  number  of  variable  factors,  temperature,  pressure  and  con- 
centration of  the  components,  which  must  be  arbitrarily  fixed 
in  order  that  the  conditions  of  the  system  may  be  perfectly 
defined."  This  will  be  better  understood  by  considering  a 
few  examples.  Take  the  case  of  liquid  water  in  contact  with 
water  vapour.  This  is  a  two-phase,  one-component  system. 
We  know  as  a  fact  of  experience  that  these  two  phases  can 
coexist  over  a  considerable  range  of  temperature,  without 
either  disappearing  or  any  new  phase  appearing.  If  we 
arbitrarily  fix  the  temperature  of  the  system  (say  25°  C.),  we 
know  also  as  a  fact  of  experience  that  the  pressure  will  take 
up  a  certain  equilibrium  value,  namely,  the  pressure  of  satu- 
rated water  vapour  at  25°  C.,  and  the  system  will  remain  in 
this  state  for  infinite  time.  On  altering  the  temperature  to 
another  value  (within  limits)  other  equilibrium  positions  will 
be  taken  up.  In  fact,  in  this  simple  case  the  temperature  of 
equilibrium  is  entirely  defined  by  one  arbitrary  variable,  the 
temperature.  That  is,  this  system  possesses  one  degree  of 


THE  PHASE  RULE  247 

freedom.  We  shall  see  later  how  this  is  predicted  by  the 
Phase  Rule.  Suppose,  however,  that  we  lower  the  tempera- 
ture of  the  system  until  ice  makes  its  appearance.  The 
system  then  consists  of  three  phases — ice,  liquid  water, 
vapour.  It  is  a  fact  of  experience  that  these  three  phases  can 
only  coexist  in  equilibrium  at  a  single  point,  i.e.  at  a  single 
value  of  temperature  and  pressure,  the  so-called  triple  point, 
which  is  approximately  o°  C.  In  this  case  we  cannot  alter  any 
variable  at  will,  for  if  we  alter,  say,  the  temperature  by  raising 
it,  the  system  will  change  in  the  sense  that  ice  will  disappear, 
and  we  are  no  longer  dealing  with  the  equilibrium  of  the 
three  phases.  Similarly,  if  we  lower  the  temperature,  the 
liquid  will  entirely  change  into  ice,  and  we  are  left  with 
the  two-phase  equilibrium,  solid — vapour.  A  system  which 
can  only  exist  at  a  single  point,  is  easily  determined  by  the 
thermodynamic  criteria,  or  equations  which  now  fix  all  the 
variables.  The  system  ice — liquid  water — vapour,  is  said  to 
possess  no  degrees  of  freedom  or  to  be  invariant.  If,  on  the 
other  hand,  we  consider  again  the  liquid  water — vapour  case, 
and  raise  the  temperature  beyond  the  critical  point,  i.e.  beyond 
the  point  at  which  the  liquid  water  can  exist  at  all,  the  system 
will  change  entirely  into  one  phase,  the  vapour,  and  this 
vapour  can  exist  as  such  under  changes  of  both  temperature 
and  pressure.  That  is,  the  system  possesses  two  degrees  of 
freedom.  Note,  we  look  upon  the  persistent  existence  of  a 
phase  as  the  experimental  evidence  for  believing  that  the 
phase  represents  an  equilibrium  or  stable  state.  In  the  above 
simple  case  we  have  not  had  to  consider  concentration  vari- 
ables, for  the  system  is  one  component  (concentration  being 
defined  as  the  ratio  of  one  component  to  another).  Hence  the 
concentration  in  each  phase  remains  constant  throughout.  Let 
us  now  take  a  case  in  which  concentration  changes  can  occur, 
namely,  the  two-component  system  H2O— NaCl.  Suppose  we 
consider  first  of  all  the  aqueous  solution  in  contact  with  vapour. 
How  many  independent  variables,  i.e.  degrees  of  freedom, 
does  such  a  system  possess?  Let  us  arbitrarily  choose  a 

N 
certain  strength  of  solution,  say  — .     Let   us   also  select  a 


248        A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

certain  temperature,  then  the  vapour  pressure  will  automati- 
cally fix  itself  and  equilibrium  will  be  maintained,  the  pressure 
being  that  of  water  vapour  (saturated)  over  the  salt  solution 

N 
of  concentration  — .     (This  vapour  pressure  we  already  know 

is  less  than  the  vapour  pressure  over  pure  water  at  the  same 
temperature.)  Before  equilibrium  could  be  fixed  it  was 
necessary  to  assign  arbitrary  values  to  the  two  variables,  con- 
centration and  temperature.  That  is,  the  two-phase  system 
considered  possesses  two  degrees  of  freedom.  Of  course  the 
actual  choice  of  variables  to  which  we  wish  to  assign  arbitrary 
values  would  not  necessarily  be  the  temperature  and  con- 
centration, although  these  are  the  most  convenient  in  the 
above  illustration.  Thus  let  us  suppose  that  we  assign  a 
certain  pressure  and  temperature  to  the  system.  Then  in 
order  that  the  system  may  realise  these  and  at  the  same  time 
be  in  equilibrium  we  must  alter  the  concentration  of  the  salt 
to  suit.  Hence  the  concentration  of  the  system  is  not  an 
arbitrary  quantity,  but  is  defined  by  the  fact  of  having  assigned 
arbitrary  values  to  the  temperature  and  pressure.  Again  we 
see  that  the  system  has  two  degrees  of  freedom  (i.e.  is 
bivariant)  although  our  choice  of  the  actual  arbitrary  variables 
differs  from  that  in  the  first  case.  In  an  exactly  analogous 
way  we  could  have  chosen,  arbitrarily,  a  certain  pressure  and 
concentration^  and  we  would  have  found  it  necessary  to  alter 
the  temperature  of  the  system  in  order  that  the  system  might 
reach  an  equilibrium  state  corresponding  to  the  arbitrarily 
chosen  temperature  and  concentration  values.  Suppose  now 
we  cause  a  new  phase  to  make  its  appearance,  say  ice.  The 
system  must  be  cooled  down,  of  course.  Suppose  we  fix  the 
concentration  of  the  solution.  Then  it  will  be  found  that 
such  a  solution  "  freezes  at  a  certain  temperature."  This  is 
the  same  thing  as  saying  that  the  system  ice — liquid  solution 
of  salt  of  fixed  concentration — vapour  can  coexist  in  equili- 
brium at  a  single  temperature  and  pressure.  Having  fixed 
the  concentration  of  the  solution,  the  equilibrium  of  the  three 
phases  is  altogether  fixed.  That  is,  the  system  possesses  one 
degree  of  freedom  (the  concentration  in  this  case).  If  we 


THE   PHASE  RULE  249 

alter  the  concentration  of  the  solution,  say,  by  increasing  it,  the 
solution  will  freeze  at  a  lower  temperature.  That  is,  once 
more  equilibrium  is  obtained,  but  the  fact  'of  altering  the 
concentration  has  caused  a  concomitant  change  in  the  tempera- 
ture and  pressure  of  the  system.  We  might  choose  any  one 
of  the  variables,  temperature  or  concentration,  but  it  will  be 
found  that  having  arbitrarily  fixed  the  value  of  one  all  others 
are  thereby  fixed  too,  i.e.  we  cannot  assign  arbitrary  values  to 
two  variables  simultaneously.  If  we  attempt  to  do  so  the  system 
will  alter  in  respect  of  the  number  of  its  phases,  i.e.  the  ice 
may  disappear,  for  example.  Having  illustrated  what  is  meant 
by  a  "  degree  of  freedom,"  we  can  now  proceed  to  the  deduc- 
tion of  the  Phase  Rule— a  generalisation  which  allows  us  to 
predict  many  of  the  facts  already  referred  to  as  having  been 
experimentally  obtained. 

Statement  of  the  Phase  Rule. 

The  Phase  Rule  states  that  a  system  consisting  of  n  com- 
ponents and  r  phases  is  capable  of  (n  —  r  -\-  2)  independent 
variations.  That  is,  the  number  of  independently  variable 
quantities  or  "degrees  of  freedom"  of  such  a  system  is 
(n  —  r-\-  2).  This  refers,  of  course,  to  the  equilibrium  state 
finally  reached.  Denoting  the  number  of  degrees  of  freedom 
by  f,  we  can  write  the  Phase  Rule  in  the  form  of  the  equa- 
tion— 

/==  u  —  r+2 

This  generalisation  holds  good  for  all  cases  in  which  electrical, 
capillary,  gravitational,  and  radiational  effects  are  absent  or 
negligible.  Before  proceeding  to  the  deduction  of  the  Phase 
Rule  it  is  necessary  to  understand  what  is  meant  by  "the 
number  of  variables  of  a  system" 

In  any  system  one  can  alter  the  temperature  and  pressure. 
These  represent  two  of  the  variables  of  the  system,  but  they 
do  not  represent  all  the  variables.  It  is  evident  that  com- 
ponents can  be  present  at  different  concentrations,  and  hence 
the  number  of  concentration  variables  must  also  be  taken  into 


250       A    SYSTEM   OF  PHYSICAL   CHEMISTRY 

account.  Let  us  fix  gur  attention  on  one  of  the  components 
only,  which  we  may  denote  by  a.  Concentration  terms  are 
essentially  ratio  terms,  say,  the  ratio  of  the  number  of  molecules 
of  the  given  component  (a)  in  a  given  phase  to  the  total 
number  of  molecules  of  all  sorts  which  go  to  make  up  the 
composition  of  the  phase.  If  there  are  n  components  in  each 
phase,  the  number  of  such  concentration  terms  is  evidently 
(n —  i)  for  each  phase.  Thus  suppose  a  phase  to  consist  of 
two  components,  i.e.  a  solution  of  common  salt  in  water.  The 
composition  of  the  phase  is  completely  determined  when  we 
know  one  ratio,  i.e.  one  concentration,  namely,  the  ratio  of 

molecules  of  salt 

—f — i —  — .      For    three    components,    two 

molecules  of  salt  and  water 

salts  and  water,  the  composition  of  the  solution   is  defined 

molecules  of  ist  salt 
when   we   know   the    ratio   of  total  molecuies  in  phase  » 

molecules  of  2nd  salt 

'     That  IS>  one  ratl° 


total  molecules  in  the  phase 
tion  term  is  sufficient  to  define  the  composition  of  a  phase 
consisting  of  two  components,  two  ratios  or  concentration 
terms  are  sufficient  to  define  the  composition  of  a  phase  con- 
taining three  components,  and  therefore  (n — i)  ratios  or 
concentration  terms  are  sufficient  to  define  the  composition  of 
a  phase  containing  n  components.  If  the  system  as  a  whole 
consists  of  r  phases  (there  being  in  each  phase  ;/  components) 
evidently  the  total  number  of  concentration  ratios  or  con- 
centration variables  is  r(n  —  i).  Hence  the  total  number  of 
variables  possessed  by  a  system  consisting  of ;/  components  in 
r  phases  is — 

r(n  —  i )  -}-  2 

Deduction  of  the  Phase  Rule  by  means  of  the  Thermodynamic 
Potential  0. 

In  the  chapter  (Part  II.,  Chap.  II.)  on  the  more  advanced 
treatment  of  thermodynamics,  it  has  been  pointed  out  that  the 
value  of  the  quantity  0}  the  thermodynamic  potential  of  a  sub- 
stance, can  be  employed  as  a  criterion  of  equilibrium,  equili- 


THE   PHASE  RULE  251 

brium  being  reached  when  0  is  a  minimum.  The  0  of  a  single 
substance  varies  with  the  temperature  and  pressure.  As  long 
as  we  keep  to  pure  substances  (i.e.  one-component  systems), 
such  as  liquid  water  in  contact  with  vapour  or  ice,  the  equili- 
brium is  determined  by  temperature  and  pressure  alone,  since 
concentration  is  necessarily  constant,  being  always  unity.  If 
the  temperature  and  pressure  of  the  liquid  phase  is  equal  to  the 
temperature  and  pressure  of  the  gaseous  or  solid  phase,  then 
we  know  that  the  0  of  each  phase  is  the  same.  When  con- 
centration terms  enter,  however,  the  0  of  a  component 
depends  on  its  concentration  in  a  phase  as  well  as  on  the 
temperature  and  pressure  of  the  system.  If  we  consider  a 
substance,  i.e.  a  component  distributed  between  two  phases, 
both  phases  being  at  the  same  temperature  and  pressure,  we 
can  say  that  equilibrium  distribution  will  be  obtained  when 
the  <P  of  the  component  in  the  first  phase  =  the  0  of  the  same 
component  in  the  second  phase.  Throughout  a  system  consist- 
ing of  many  phases  (say  r  phases),  each  component  being  pre- 
sent to  a  greater  or  smaller  extent  in  every  phase,  equilibrium 
with  respect  to  any  one  of  the  n  components  is  reached  when 
the  0  of  that  component  has  the  same  value  in  every  phase. 
Consider  such  a  system  consisting  of  n  components  in  r 
phases,  and  let  us  fix  our  attention  on  one  component  which 
we  can  denote  by  a.  Suppose  an  infinitely  small  quantity 
dma  of  component  a  is  transferred  from  the  first  phase  to  the 
second  phase.  The  decrease  in  the  0  of  the  component  a  in 
the  first  phase,  which  we  can  denote  by  0ai  is  given  by  the 
expression— 


The  increase  in  the  0  of  the  same  component  in  the  second 
phase  is  given  by — 

•  i^n 
r  d»fa 

Now  if  the  system  is  in  equilibrium  as  a  whole  the  component 
a  must  have  distributed  itself  in  equilibrium  throughout  all  the 


252        A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

phases,  and  therefore^  between  the  first  and  second  phases. 
But  the  criterion  of  the  equilibrium  distribution  of  the  com- 
ponent a  (the  temperature  and  pressure  of  which  we  have 
kept  constant,  having  simply  made  a  virtual  variation  in  the 
concentration)  is  that  0a  shall  have  remained  unchanged ;  that 
is  d0a,  =  o.  But — 


r»r  — 

~n  -  "  —     o  ~ 

dma        dm* 


Note  these  are  partial  differentials,  temperature  and  pressure 
being  assumed  constant.  Similarly  considering  the  equili- 
brium distribution  of  the  same  component  a  between  the 
second  and  third  phases,  we  would  obtain  another  equation 
for  the  equilibrium,  viz.  — 


dma          dma 

Thus  the  equilibrium  of  one  component  distributed  between 
three  phases  (I.,  II.,  and  III.)  is  denned  by  two  equations. 
Hence  the  number  of  equations  dealing  with  the  distribution 
of  one  component  amongst  r  phases  would  be  (r  —  i).  If  we 
now  consider  each  component  in  turn,  distributed  throughout 
the  r  phases,  the  total  number  of  thermodynamic  equations 
would  be  n(r  —  i). 

The  number  of  variables  possessed  by  a  system  containing 
n  components  in  r  phases  has  been  shown  in  the  preceding 
section  to  be  — 


Hence  the  number  of  degrees  of  freedom,/  which  represent  the 
number  of  variables  for  which  there  are  no  equations  is 
evidently  the  difference  between  the  total  variables  and  the 
total  equations,  viz.  — 

r(n—  i}  +  2  —  n(r—  i) 
or  /=  n  —  r-\-2 


THE   PHASE  RULE  253 

Note  on  the  Preceding  Deduction  of  the  Phase  Rule. 

Attempts  have  been  made  from  time  to  time  to  "  deduce  " 
the  Rule  by  employing  somewhat  simpler  conceptions  than 
that  of  the  thermodynamic  potential.  The  following,  for 
example,  has  been  suggested  by  J.  A.  Muller  (Comptes  Rendus^ 
146,  p.  866,  1908).  Equilibrium  of  a  component  throughout 
a  system  (of  n  components  in  r  phases)  is  obtained  when  the 
mass  m  of  the  component  passing  per  second  from  Phase  I.  to 
Phase  II.  is  equal  to  the  mass  m'  of  the  same  component 
passing  per  second  from  Phase  II.  to  Phase  I.  That  is,  as  a 
kinetic  criterion  of  equilibrium  we  have  m  =  m' .  Considering 
the  same  component  throughout  all  the  phases  we  obtain 
(/- —  i)  similar  equations,  and  hence  for  n  components  we 
obtain  «(;• —  i)  equations.  The  total  number  of  variables 
being  r(n  —  i)  +  2,  and  the  number  of  degrees  of  freedom/— 

f  =  r(n  —  i)  +  2  —  n(r  —  i)  =  ;/  —  r  -f-  2 

which  is  the  Phase  Rule.  It  will  be  evident,  however,  that 
such  a  method  of  prpcedure  is  open  to  serious  doubt.  Prof. 
W.  B.  Morton  *  suggests  the  following  criticism.  "  In  the 
proof  of  the  Phase  Rule  the  essential  thing  is  that  when  two 
phases  are  in  equilibrium,  in  contact,  then  the  condition  of 
equilibrium  is  expressed  by  the  equality  of  some  function  of 
the  variables  for  one  phase,  to  some  function  of  those  same 
variables  for  the  other  phase.  That  this  function  is  0  is 
immaterial  to  the  proof,  which  is  concerned  only  with  the 
number  of  the  equations  and  not  with  their  particular  form. 
Of  course  one  admits  that  equilibrium  can  be  expressed  by 
some  equation  connecting  the  variables  of  the  two  phases 
concerned.  But  this  is  not  enough — for  there  might  then  be 
as  many  equations  as  there  are  pairs  of  phases.  The  equa- 
tions might  be  written  thus — 

F(I,  II)  =  o,  F(I,  III)  -  o,  F(II,  III)  =  o 

and  so  on,  where  F  stands  for  some  function  connecting  the 
phases.  The  number  of  equations  of  this  type  reckoned  for 

1  Private  communication. 


254       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

r  phases  is  obviously  lr(r —  i)  in  all  (instead  of  the  number 
(r —  i)  which  appears  in  the  Phase  Rule  deduction).  It  is 
essential  that  F(I,  II)  =  o  should  be  expressible  in  the  form 
<£j  =  0Ut  i.e.  that  it  should  be  possible  to  separate  the  vari- 
ables of  one  phase  from  those  of  the  other,  and  that  the 
collection  of  variables  in  Phase  I.  indicated  by  <2>i  should  be 
independent  of  the  other  phase  coupled  with  it  in  the  equa- 
tion. Then  and  only  then,  do  the  equations  reduce  to  the 
set — 

0i  =  0U  =  (2>m,  etc.,  or  (r  —  i)  in  number." 

The  kinetic  deduction  referred  to  above  appears  to  make  the 
assumption  that  the  mass  of  a  component  leaving  a  phase  per 
second  is  independent  of  the  other  phase  into  which  it  passes, 
e.g.  that  Phase  II.  loses  per  second  as  much  of  each  com- 
ponent into  Phase  I.  as  into  Phase  III.  This  is  by  no  means 
obvious.  We  have  only  to  think  of  common  salt  diffusing  on 
the  one  side  into  an  acetone  phase  in  which  its  solubility  is 
extremely  small,  and  on  the  other  into  an  aqueous  phase,  to 
see  that  the  quantities  passing  out  into  the  two  phases  per 
second  will  in  all  probability  be  very  different.  Granting, 
however,  that  the  same  mass  per  second  does  travel  from 
II.  to  III.  as  from  II.  to  I.,  then  m  is  the  mass  of  the  component 
leaving  Phase  I.  per  second,  and  this  may  be  regarded  as  a 
function  of  the  variables  in  Phase  I.  only,  and  corresponds 
therefore  to  the  "  some  function "  mentioned  above.  The 
deduction  proceeds  then,  m  taking  the  place  of  0  (m  might 
be  regarded  perhaps  as  a  sort  of  kinetic  measure  of  0).  It 
will  be  clear,  however,  that  the  kinetic  method  of  the  kind 
employed  by  Muller  is  open  to  much  doubt. 


CLASSIFICATION  OF  (HETEROGENEOUS)  SYSTEMS  BY  MEANS 
OF  THE  PHASE  RULE. 

The  most  convenient  method  of  grouping  is  according  to 
the  number  of  components  forming  the  system. 


HETEROGENEOUS  SYSTEMS  255 

One-Component  Systems. 
Typical  instances  are — 


Component. 


Possible  phases. 


More  than  one  solid — liquid — vapour. 


Water 

Sulphur 

Tin „  „  „ 


Two-Component  Systems. 


Components. 


Possible  phases. 


CaO,  C0fl      .      .  %. 
CuSO4,  H2O  .      .      . 


FeCl3, 


Solids  :  (CaCO3,  CaO)— gas  (CO0). 

Solids  :  CuSO4  .  5H2O,  CuS"O43H2O, 
CuSO4,  ice — liquid  \  solutions — wafer- 
vapour. 

Solids  :     FeCl3     anhydrous     and     several 


hydrates,  ice — liquid  :  solutions — water- 
vapour. 

Na2SO4,  H2O  .  '  Solids  :  anhydrous  salt,  hepta  and  deka 

hydrate,  ice — liquid  '.  solutions — water- 
vapour. 

Fe,  C Solids  :  alloys  (steels) — liquid — vapour. 


Three- Component  Systems. 


Components.  Possible  phases. 


Two   salts   with   a   common    \ 

ion — H2O 

NaCl — KC1 — H,O    .      .      .     >•   Several  solids — solution — vapour. 
KC1— MgClo— H20  .      .      . 
NaO  -(CH3CO)20— H20  '     ) 


Four-Component  Systems. 

Components :  Two  salts  without  a  common  ion  (so-called 
reciprocal  salt  pairs  such  as  sodium  chloride  and  ammonium 
bicarbonate)  and  water  in  addition.     This  particular  system  is 
1  Dunningham,  Trans.  Chem.  See.,  101,  431,  1912. 


256        A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

that  occurring   in   the  ammonia   soda  or   Solvay  process  of 
formation  of  sodium  <mrbonate. 

Possible  Phases  :  several  solids—  solution—  vapour 


ONE-COMPONENT  SYSTEMS  :   COMPONENT  H2O. 

We  shall  consider  the  conditions  of  equilibrium  in  some  of 
the  systems  to  which  this  component  can  give  rise.  Take  the 
system  — 

Liquid  water—  water  vapour  in  contact. 

This  system  consists  of  two  phases.     Applying  the  Phase  Rule, 
since  n  =  i,  and  r  =  2,  we  get  — 

f=n  —  /•  -f-  2  =  i  —  2  -J-  2  :=  i 

This  system  possesses  one  degree  of  freedom,  i.e.  a  univariant 
system.     That  is,  if  we  arbitrarily  fix  the  temperature,  say,  then 


the  system  as  a  whole  will  assume  a  certain  equilibrium  state. 
By  choosing  a  series  of  temperature  values  we  obtain  a  corre- 
sponding series  of  equilibrium  states.  This  can  be  indicated 
on  a  temperature  pressure  diagram  by  means  of  the  usual 
vapour  pressure  curve  (cf.  Fig.  73).  On  a  diagram  such  as 


ONE-COMPONENT  SYSTEMS  257 

this  an  invariant  system  (i.e.  a  system  without  any  degrees  of 
freedom)  is  represented  by  a  point.  A  univariant  system  is 
represented  by  a  line,  e.g.  the  liquid  water — vapour  curve  OA. 
Bivariant  systems,  or  systems  having  two  degrees  of  freedom 
are  represented  by  areas.  The  upper  limit  A  is  the  critical 
point,  viz.  364-3°,  and  194*6  atmospheres  pressure. 

System :  Solid  water  (ice] — vapour. 

This  being  also  a  univariant  system  must  be  represented 
by  a  line,  viz.  OB.  This  has  the  same  significance  for  the  solid — 
vapour  system  as  the  vapour  pressure  curve  had  for  the 
liquid— vapour  case.  In  fact,  OB  is  a  vapour  pressure  curve, 
or,  as  it  is  more  usually  called  when  solid  gives  rise  to  vapour, 
a  sublimation  curve.  Since  ice  melts  at  o°  C.  under  atmo- 
spheric pressure,  or  at  -f-  0*007°  C.  when  under  the  pressure 
of  its  saturated  vapour,  viz.  4*579  mm.  Hg,  this  latter  tempera- 
ture must  mark  the  upper  limit  of  the  sublimation  curve  on  a 
temperature-pressure  diagram.  Existence  of  ice,  i.e.  ordinary 
ice^  above  +  0-007°  C.  has  never  been  observed.  On  the 
other  hand,  it  is  possible  to  super-cool  liquid  water  below 
0*007°  C.  (the  temperature  at  which  liquid  water  under  its 
own  vapour  pressure  should  solidify).  This  meta  stable  state 
of  the  liquid — vapour  system,  is  represented  in  the  diagram 
by  the  dotted  line  OA',  which  it  is  important  to  note  is  an 
unbroken  continuation  of  the  stable  vapour  pressure  curve  OA. 
Note  particularly  that  OA'  lies  above  OB.  That  is,  at  any 
given  temperature  below  the  point  O,  the  vapour  pressure 
over  the  super-cooled  water,  which  is  the  unstable  phase,  is 
greater  than  the  vapour  pressure  over  the  stable  phase,  solid 
ice.  We  have  already  seen,  on  thermodynamic  grounds,  that 
this  must  be  so  (Part  II.,  Chap.  IV.). 

The  System;  Liquid  water — ice  (ordinary  form). 

A  one-component  system  consisting  of  these  two  phases 
is  also  bivariant.     The  series  of  equilibrium  states,  correspond- 
ing, say,  to  a  series  of  pressure  values,  is  represented  by  the 
T.P.C, — n.  s 


258        A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

line  OC.  The  line  QP  is  the  freezing-point  curve.  The  upper 
limit  as  far  as  temperature  is  concerned  is  -f-  0*007°  C.  since 
ordinary  ice  cannot  exist  above  this  temperature.  By  increas- 
ing the  pressure  on  ice  alone,  we  know  that  it  will  melt.  This 
means  that  if  we  have  ice  and  water  together,  under  a  certain 
pressure,  say  the  pressure  of  saturated  vapour,  there  will  be 
an  equilibrium  at  O,  and  on  arbitrarily  increasing  the  pressure 
it  will  be  necessary  to  lower  the  temperature  of  the  system  in 
order  to  keep  the  ice  as  a  permanent  part  of  it.  On  the 
basis  of  the  Le  Chatelier-Braun  principle,  the  change  of 
freezing  temperature  (i.e.  equilibrium  temperature)  with  pres- 
sure is  extremely  small,  as  one  would  expect  from  the  small 
volume  change  that  accompanies  fusion,  and  hence  the  line 
OC  is  very  steep. 

The  System  :   Ordinary  ice — liquid  water — vapour. 

This  system  consists  of  three  phases,  and  since  there  is 
only  one  component  /=  o,  i.e.  the  system  is  invariant  or  has 
no  degrees  of  freedom.  This  means  that  we  cannot  fix  any 
condition  (such  as  temperature  and  pressure)  arbitrarily.  This 
system  can  only  permanently  exist  at  a  single  point  on  the 
temperature-pressure  diagram.  This  equilibrium  point  is  O, 
which  corresponds  to  the  conditions — • 

T  —  +  0*0076°  C.,  P  =  4*579  mm.  mercury 

If  we  attempt  to  alter  either  T  or  P,  one  of  the  phases  will 
disappear.  Thus  on  raising  the  temperature  to  the  slightest 
extent  the  ice  phase  will  vanish,  and  the  system  will  again  be 
in  equilibrium  on  the  OA  curve,  infinitely  close  to  O.  On 
lowering  the  temperature  the  vapour  and  the  liquid  phase  will 
both  tend  to  disappear.  Which  will  disappear  first  depends 
on  the  absolute  amounts  of  each.  If  the  vapour  disappears 
first  the  system  will  again  cease  changing,  i.e.  will  come  into 
an  equilibrium  position  on  the  OC  curve  at  a  point  infinitely 
close  to  O.  If  the  liquid  phase  disappears1  first  (i.e.  becomes 

1  We  are  here  leaving  out  of  account  meta-stable  conditions,  which  do 
not  represent  true  equilibrium  states. 


ONE-COMPONENT  SYSTEMS  259 

solid)  the  system  will  be  on  the  curve  OB.     The  point  O  at 
which  three  phases  are  in  equilibrium  is  called  a  triple  point. 

The  Hypothetical  Case :  Two  forms  of  ice  * — liquid  water — 
vapour. 

This  being  a  one-component  system  n  =  i 

There  are  four  phases  r  =  4 

.'.  f=  n  —  r  -j-  2  =  i  —  4  -|-  2  =  —  i 

Such  a  system  would  possess  a  negative  value  for  the  degrees 
of  freedom.  To  have  less  than  zero  degrees  of  freedom  is 
unrealisable.  We  can  therefore  infer,  on  the  basis  of  the 
Phase  Rule,  that  it  is  an  impossibility  to  have  a  stable  system 
consisting  of  only  one  component  in  more  than  three  phases. 
If  such  a  system  be  set  up,  one  of  the  phases  will  commence 
to  disappear  no  matter  what  values  we  give  to  the  temperature 
and  pressure.  This  is  a  striking  instance  of  the  usefulness  of 
the  Phase  Rule  as  a  guide  to  the  chemical  behaviour  of 
heterogeneous  systems. 

Various  Forms  of  Ice  and  the  Corresponding  Systems  which 
can  be  formed  thereby. 

So  far  we  have  spoken  of  ice  as  "  ordinary  ice,"  as  though 
there  were  only  one  crystalline  form  or  modification  of  water 
in  the  solid  state.  Tammann,2  however,  has  found  three 
crystalline  forms,  i.e.  three  different  phases  in  which  solid 
H2O  can  exist.  Ordinary  ice  is  called  by  him,  ice  I.,  the 
other  varies  being,  ice  II.,  and  ice  III.  These  latter  varieties 
can  be  reached  at  low  temperatures  and  very  high  pressures. 
According  to  the  Phase  Rule  one  may  expect  to  have  a 
number  of  systems  in  addition  to  those  already  discussed. 
Of  these  Tammann  found  the  following  Invariant  Systems  : 

liquid  water — ice  I. — ice  III. 
ice  I. — ice  II. — ice  III. 

1  See  later  Tammann  and  Bridgman's  work  on  the  various  forms  of  ice. 

2  Tammann,  Annal  der  Physik.,  [4]  2,  1424,  1900. 


260        A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

Each  of  these  systems  exist  in  equilibrium  only  at  one  point 
characteristic  of  the  phases  present.  Hence  beside  the  point 
0  (Fig.  73),  Tammann's  results  show  the  existence  of  two 
other  triple  points. 

Univariant  Systems : 

liquid  water — ice  III. 
ice    I.— ice  II. 
ice    I. — ice  III. 
ice  II.— ice  III. 

These  would  be  represented  by  lines  in  a  T.P.  diagram. 
Bivariant  Systems. 

ice  II. 
ice  III. 

Each  of  these  bivariant  systems  is  represented  by  an  area 
on  the  T.P.  diagram,  i.e.  we  can  alter  both  temperature  and 
pressure  simultaneously  within  limits,  without  causing  the  phase 
to  disappear.  The  work  of  Tammann  has  recently  been  ex- 
tended by  the  investigation  of  the  H2O  system  at  low  tempera- 
tures and  very  high  pressures  by  P.  W.  Bridgman  (Proc.  Ainer. 
Acad.\  47,44i,  1912).  Tammann,  working  up  to  3500  kilograms 
per  square  centimeter  pressure,  had  shown  the  existence  of 
three  different  forms  of  ice ;  Bridgman,  by  working  up  to  20,500 
kilos  per  square  centimeter,  has  succeeded  in  demonstrating  the 
existence  of  yet  two  other  stable  forms,  making  five  definite 
crystalline  varieties  of  ice.  All  these  forms,  with  the  exception 
of  ordinary  ice  (ice  I.)  are  denser  than  that  of  water.  Bridgman 
was  able  to  locate  five  of  the  six1  possible  triple  points  (three 
phases  in  equilibrium)  and  ten  out  of  the  eleven  possible  tran- 
sition lines  (two  phases  in  equilibrium)  have  been  followed. 
The  sixth  triple  point  and  the  eleventh  equilibrium  line  lie  at 
temperatures  so  low  and  pressures  so  high  that  the  slowness 
of  the  reaction,  i.e.  the  rate  of  transformation  of  one  variety 
into  another  makes  them  practically  impossible  to  determine. 
The  passage  of  one  phase  into  another  we  speak  of  as 
"  transition."  The  lines  OA,  OB,  OC,  in  the  figure  73,  are 

1  Six  in  addition  to  the  triple  point  O  (Fig.  73)  where  ice  I. — liquid — 
vapour  coexist  in  equilibrium. 


ONE-COMPONENT  SYSTEMS 


261 


transition  lines.  The  triple  point,  such  as  O,  is  called  the 
"transition  point."  The  method  of  determining  the  actual 
existence  of  new  substances,  such  as  the  new  crystalline  forms 
of  ice,  is  to  try  and  map  out  on  a  T  and  P  diagram,  the 
direction  and  length  of  transition  lines  and  determine  the 


-80°    -60°  -4-0°    -20°     0°       20°      40°      60°      80° 

Temperature 

Equilibrium  Diagram  between  liquid 
water  &  the  5  sol  id  modifications  of  ice 
FIG.  74. 

location  of  transition  points.  In  this  way  we  can  visualise 
the  range  of  stable  existence  as  regards  T  and  P  of  each  phase 
in  contact  with  other  phases.  Such  a  diagram  given  in  the 
figure  (Fig.  74)  has  been  mapped  out  by  Bridgman,  showing 
the  equilibrium  relations  of  the  five  different  forms  of  ice  and 
the  liquid.  The  various  forms  of  ice  are  denoted  by  the 
symbols  ice  I.,  II.,  III.,  V.,  and  VI.,  respectively.  The  new 


262        A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

forms  discovered  by  JSridgman  are  ice  V.  and  ice  VI.  The 
absence  of  the  term  ice  IV.  is  due  to  the  fact  that  although 
Tammann  obtained  some  evidence  (Zeit.  physik.  Chem^  72,  609, 
1910)  for  such  a  modification,  there  still  remains  considerable 
doubt  in  regards  to  its  existence.  If  later  it  should  be  found, 
space  is  left  for  it  on  the  above  scheme  of  nomenclature.  (For 
a  discussion  of  ice  IV.,  see  Bridgman's  paper,  /.<:.,  pp.  527  seq.} 
It  will,  perhaps,  make  the  diagram  more  easily  understood 
if  we  imagine  ourselves  following  out  the  behavour  of  the  H2O 
system,  starting  from  the  ordinary  triple  point  O.  The  observer 
is  supposed  to  be  able  to  distinguish  by  sight  simply  the 
different  forms  of  ice,  even  though  the  actual  enclosing  vessel 
is  necessarily  hard  steel  to  withstand  the  high  pressure  im- 
pressed upon  the  system  by  means  of  a  steel  plunger.  If  the 
pressure  at  O  be  increased,  the  vapour  phase  may  be  caused 
to  disappear,  and  we  are  left  with  the  liquid — ice  I.  system. 
By  continuously  increasing  the  pressure  and  suitably  lowering 
the  temperature  we  can  keep  these  two  phases  in  equilibrium, 
i.e.  we  can  pass  through  a  consecutive  series  of  equilibrium 
points  represented  by  the  curve  OC.  This  curve  is  a  con- 
tinuous one  having  the  shape  indicated  until  we  reach  the 
pressure  of  2115  kilos  per  square  centimeter  and  the  tempera 
ture  —  22°  C.,  at  which  point  there  is  a  sharp  discontinuity 
indicated  by  the  point  C.  The  system  will  now  be  "  seen  "  to 
consist  of  three  phases,  ice  I. — ice  III.  and  liquid.  Point  C  is 
therefore  a  triple  point.  If  we  now  further  increase  the 
pressure,  and  at  the  same  time  raise  the  temperature  very 
slightly,  ice  I.  will  disappear  and  we  are  left  with  ice  III.  and 
liquid.  By  further  increasing  the  pressure  these  two  phases 
can  be  kept  in  equilibrium  by  suitably  raising  the  temperature, 
and  in  this  way  we  can  pass  along  the  line  CD  with  con- 
tinuously increasing  pressure.  At  D  the  CD  line  is  broken 
sharply  and  another  solid  phase  makes  its  appearance,  namely 
ice  V.,  point  D  being  the  triple  point  corresponding  to  the 
co-existence  of  ice  III.,  ice  V.  and  liquid.  The  pressure  is 
3530  kilos  per  square  centimeter  and  the  temperature  — 17°  C. 
On  still  further  increasing  the  pressure  and  raising  the  tem- 
perature again  slightly,  ice  III.  can  be  made  to  disappear,  and 


ONE-COMPONENT  SYSTEMS  263 

we  may  pass  along  the  curve  DE  which  represents  the  equi- 
librium line  of  ice  V.  and  liquid.  At  the  point  E  (pressure 
6380  kilos  per  square  centimeter,  temperature  -|-o'i6°  C.)  the 
curve  DE  is  broken,  the  phase  ice  VI.  making  its  appearance, 
the  point  E  being  the  triple  point  for  the  three  phases,  ice  V., 
ice  VI.,  and  liquid.  On  increasing  the  pressure  it  is  possible 
to  follow  the  equilibrium  stages  of  ice  VI.  in  contact  with 
liquid.  This  curve  has  been  traced  without  showing  any  break 
up  to  pressures  of  the  order  of  20,000  kilos  per  square  centi- 
meter, the  limit  of  Bridgman's  pressure  apparatus,  and  tem- 
perature about  -{-  80°  C.,  without  any  new  phase  making  its 
appearance.  These  results  show  that  it  is  actually  possible  to 
have  H2O  in  the  solid  form  in  equilibrium  with  liquid  at  a 
temperature  as  high  as  -j-8o°  C.  The  solid  here  is,  however, 
not  ordinary  ice  but  a  new  modification  ice  VI. 

Now  let  us  return  to  the  triple  point  C.  Let  us  increase 
the  pressure,  only  to  an  extremely  slight  extent,  and  let  us 
lower  the  temperature  to  a  suitable  extent  (instead  of  raising 
it).  The  liquid  phase  will  now  disappear  and  the  system  will, 
consist  of  ice  I.  and  ice  III.,  the  equilibrium  between  these 
two  states  being  followed  by  lowering  the  temperature  and 
suitably  adjusting  the  pressure.  In  this  way  we  pass  along  a 
continuous  curve  CF  which  shows  a  break  (i.e.  a  sudden  change 
in  direction)  at  F,  at  which  point  a  new  solid  form  is  produced, 
namely  ice  II.  F  is  the  triple  point,  ice  III.,  ice  II.,  ice  I. 
(pressure  =  2170  kilos  per  square  centimeter,  temperature 
— 34*7°  C.).  By  still  further  lowering  the  temperature,  ice  III. 
can  be  caused  to  disappear,  the  remaining  phases,  ice  II.  and 
ice  I.,  being  kept  in  equilibrium  by  slightly  lowering  the 
pressure,  i.e.  we  pass  along  the  line  FR.  Note  that  the  line 
FR'  is  the  metastable  equilibrium  line  between  ice  I.  and  ice 
III.  That  is,  we  have  passed  the  point  F  at  which  ice  II. 
should  appear,  but  owing  to  the  slowness  of  reaction  in  solid 
phases,  especially  at  low  temperatures,  we  have  super-cooling 
of  the  ice  III.  phase.  The  prolongation  of  lines  through  the 
other  triple  points  represent  the  same  phenomena  also  taking 
place.  This  phenomena  of  "  suspended  transformation  "  is  of 
quite  frequent  occurrence,  and,  as  one  might  expect,  it  greatly 


264       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

increases  the  difficulty  of  accurately  determining  the  true 
transition  point  (in  the  present  case  a  triple  point).  In  the 
case  of  two  solid  phases,  the  transition  point  can  be  over- 
stepped in  both  directions.  In  the  case  of  the  transition  point 
of  a  solid  and  liquid  together  the  liquid  can  be  super-cooled 
easily,  but  no  case  yet  has  been  recorded  in  which  the  solid 
has  been  heated  above  the  triple  point  without  passing  into 
the  liquid  state.  Suspended  transformation  is  therefore  limited 
to  one  direction.  This  is  the  only  distinction  between  a 
solid — solid  transition  point  and  a  solid — liquid  melting  point. 
Again,  if  we  imagine  ourselves  at  the  point  F  and  raise  the 
pressure,  we  can  cause  ice  I.  to  disappear  and  ice  II.  and 
ice  III.  to  remain  in  equilibrium  by  raising  the  temperature. 
In  this  way  we  pass  from  F  to  S,  at  which  point  ice  V.  makes 
its  appearance  (pressure  3510  kilos  per  square  centimeter,  tem- 
perature —  24*3°  C.),  as  on  raising  the  temperature  we  can 
cause  ice  II.  to  disappear,  and  by  suitably  increasing  the 
pressure  to  a  very  slight  extent  we  can  pass  along  SD  to  the 
•triple  point  D  already  described.  The  following  table  sum- 
marises the  T.P.  data  already  given  for  all  the  triple  points 
of  the  H2O  system  which  have  been  experimentally  realised  : — 


Phases. 

Position  on 
diagram. 

Temperature. 

Pressure. 

ice  I.  —  liquid—  vapour 
ice  I.  —  liquid  —  ice  III. 

o 

C 

+  o-oo76°C. 

-22°C. 

4  '579  mm.  Hg 
2115  kilos/cm2. 

ice  III.  —  liquid  —  ice  V. 

D 

-i7°C. 

3530 

ice  V.  —  liquid  —  ice  VI. 

E 

+  o-i6°C.        6380 

ice  I.—  ice  II.  —  ice  III. 

F 

-34'7°C.        2170 

ice  II.—  ice  III.—  ice  V.           S 

! 

-24-3°C. 

35io 

It  will  be  observed  that  the  field  of  existence  of  ice  II.  is 
bounded  by  ice  I.,  ice  III.,  and  ice  V.  It  is  impossible  to  get 
ice  II.  in  contact  equilibrium  with  liquid  under  any  tempera- 
ture or  pressure  values  whatsoever.  It  is  thus  impossible  to 
predict  (apart  from  experiment)  as  to  how  many  triple  points 
a  given  number  of  phases  may  give  rise.  We  must  find  out 
by  experiment  something  about  the  slopes  of  the  transition 
lines  between  pairs  of  phases  in  the  case  of  one-component 


ONE-COMPONENT  SYSTEMS  265 

systems  such  as  H2O,  before  we  are  able  to  state  the  probable 
sets  of  triple  points.  If  we  are  able  to  measure  the  specific 
volumes  of  two  phases  separately  we  can  foretell  by  applying 
the  Le  Chatelier-Braun  principle  what  will  be  the  direction  of 
the  slope  of  the  equilibrium  curve  on  increasing  the  pressure, 
for  under  these  conditions,  according  to  this  principle,  the 
system  will  tend  to  transform  itself  into  that  occupying  the 
least  volume.  To  prevent,  therefore,  the  total  disappearance 
of  one  of  the  phases  we  have  to  alter  some  other  condition  of 
the  system,  i.e.  the  temperature.  It  will  be  observed  that  in 
the  case  of  the  forms  ice  III.,  ice  V.,  and  ice  VI.,  increase  of 
pressure  raises  the  freezing  point,  i.e.  the  behaviour  is  the 
reverse  of  ice  I.  From  the  Le  Chatelier-Braun  principle  we 
see  that  this  is  due  to  the  fact  that  the  density  of  these  other 
forms  of  ice  is  greater  than  that  of  the  liquid  in  equilibrium. 
The  phenomenon  of"  regelation  "  can  occur  therefore  only  with 
ice  I.  Ice  II.  does  not  melt,  i.e.  ice  II.  cannot  be  brought  into 
equilibrium  with  the  liquid  form.  Starting  with  some  ice  II., 
on  raising  the  temperature  we  transform  it  into  ice  III.,  unless 
indeed  the  temperature  has  been  raised  too  quickly  and  we 
obtain  super-heated  ice  II.,  which  might  conceivably  "  melt." 
The  system,  however,  would  not  be  an  equilibrium  one. 

Of  course  in  following  out  the  diagram,  as  has  been  done 
in  the  preceding  pages,  we  are  obviously  carrying  out  a  purely 
imaginary  process.  The  actual  slopes  of  the  lines,  and  the 
discontinuities  indicating  the  points  at  which  new  phases 
appear,  are  determined  by  observing  the  change  in  some 
physical  property  of  the  system  (such  as  the  volume}  under 
varying  conditions  of  temperature  and  pressure.  Bridgman, 
following  Tamman's  method,  determined  what  was  virtually 
the  compressibility  of  the  H2O  system  under  various  condi- 
tions, the  system  being  enclosed  in  a  thick-walled  steel 
piezometer.  In  other  cases  other  properties  may  be  employed. 
These  will  be  mentioned  in  connection  with  the  particular 
system  followed.  The  more  important  may  be  summarised 
here : 

1.  Measurement  of  change  in  volume  by  means  of  a  dilato- 
meter  or  piezometer. 


266        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

II.  Measurement , of    vapour    pressure    (Bremer-Frowein 
tensimeter). 

III.  Measurement  of  solubility. 

IV.  Thermo-analytical  method. 

V.  Optical  method. 

VI.  Electrical  methods  (measurements  of  conductivity  and 
electromotive  force). 

VII.  Measurement   of  viscosity  (Dunstan   and   Langton, 
Trans.  Chem.  Soc.,  101,  418,  1912). 

An  account  of  most  of  these  methods  is  given  in  the 
appendix  to  Findlay's  Phase  Rule. 

ONE-COMPONENT  SYSTEMS  (continued). — COMPONENT  : 
SULPHUR  (S)n. 

It  is  a  fact  familiar  to  every  chemist  that  sulphur  can  exist 
in  two  crystalline  modifications,  rhombic  and  monoclinic,  as 
well  as  in  the  liquid  and  gaseous  forms.  The  conditions  of 
equilibrium  of  the  various  phases  of  this  one-component 
system,  can  be  represented  by  the  usual  P.T.  diagram  given 
in  the  figure  (Fig.  75).  In  this  case,  as  in  the  last,  the 
equilibrium  lines  and  points  have  been  determined  by 
measurements  of  the  change  of  volume,  with  changes  in  tem- 
perature and  pressure. 

Experiment  has  shown  that  if  we  start  with  solid  rhombic 
sulphur,  on  heating  it  rapidly  it  melts  at  a  temperature  ii4°C. 
(point  H).  If,  however,  we  keep  rhombic  sulphur  heated  at 
any  temperature  between  96°  C.  and  114°  C.  it  will  be  found 
to  have  become  transformed  into  monoclinic  sulphur,  which 
will  now  melt  at  120°  C.  This  rather  anomalous  behaviour  is 
easily  explained  when  we  study  the  system  systematically, 
from  the  standpoint  of  the  Phase  Rule.  Thus  Reicher  showed 
that  at  a  temperature  95*5°  C.,  and  under  a  certain  pressure, 
namely,  the  pressure  of  the  saturated  sulphur  vapour,  the 
rhombic  form  passes  into  the  monoclinic.  This  is  represented 
by  the  point  A.  This  is  a  triple  point  corresponding  to  the 
stable  coexistence  of  the  three  phases,  solid  rhombic — solid 
monoclinic — vapour.  According  to  the  Phase  Rule,  since  the 


ONE-COMPONENT  SYSTEMS 


267 


system  only  consists  of  one  component  we  cannot  get  more 
than  three  phases  to  remain  together  in  equilibrium.  If  at  the 
point  A  we  compress  the  system  until  the  vapour  phase  entirely 
disappears  it  will  be  found  that  on  further  increasing  the 
pressure  it  is  necessary  to  raise  the  temperature  in  order  to 
keep  both  rhombic  and  monoclinic  present  together.  We  thus 


I 


95-5° 


120°  131° 


Temperature 


Equilibrium  diagram  of  Sulphur 
Two  solid  phases  -liquid-  vapour. 

FIG.  75. 

pass  along  the  line  AB.  When  the  temperature  has  reached 
the  value  151°  C,  and  the  pressure  1288  atmospheres,  liquid 
sulphur  may  be  brought  into  contact  with  the  two  solid  phases 
and  the  system  is  in  equilibrium.  This  is  the  triple  point  B, 
at  which  monoclinic,  rhombic,  and  liquid  can  coexist.  This 
is  taken  as  the  true  melting  point  of  rhombic  sulphur.  If  we 
still  further  increase  the  pressure,  the  monoclinic  form  will 
entirely  disappear,  and  we  can  keep  rhombic  and  liquid  present 


268        A    SYSTEM   OF  PHYSICAL   CHEMISTRY 

together  by  suitably  raising  the  temperature.  In  this  way  we 
pass  along  the  line  BE.  Of  course  each  point  on  this  line  is 
a  melting  point  of  rhombic  under  the  given  pressure,  The 
point  B  corresponds  to  the  lowest  temperature  and  pressure  at 
which  rhombic  will  remain  stable  in  the  presence  of  liquid. 
Again  returning  to  the  point  A,  if  we  raised  the  temperature 
very  slowly  without  altering  the  pressure,  it  will  be  found  that 
rhombic  disappears$  and  we  are  left  with  monoclinic  in  the 
presence  of  vapour.  The  point  A  is  therefore  the  "  transition 
point  "  of  rhombic  into  monoclinic.  To  keep  the  monoclinic 
and  vapour  permanently  present  together,  as  we  raise  the 
temperature  the  pressure  must  likewise  be  raised.  The 
system  does  this  automatically  by  the  increase  in  its  vapour 
pressure.  We  thus  pass  along  AC.  At  the  point  C  the  mono- 
clinic  melts.  At  C  therefore  we  have  another  triple  point, 
monoclinic — liquid — vapour.  This  is  the  temperature  120°  C., 
the  true  melting  point  of  monoclinic  sulphur.  On  further 
raising  the  temperature  the  monoclinic  disappears  and  we  pass 
along  CF,  which  is  the  "  vapour  pressure  curve  "  or  equilibrium 
curve  of  liquid  sulphur  in  contact  with  vapour.  If  at  the 
point  C  we  had  compressed  the  system  so  as  to  make  the 
vapour  phase  disappear,  and  then  had  further  increased 
the  pressure,  it  would  be  found  necessary  to  raise  the  tempera- 
ture in  order  to  keep  monoclinic  and  liquid  present  together. 
By  successively  increasing  the  pressure  and  temperature  we 
can  pass  along  the  line  CB,  which  represents  the  melting  point 
of  monoclinic  under  various  pressures.  At  B  we  can  add 
some  rhombic,  and  the  three  phases  will  be  found  to  be  in 
equilibrium  as  already  mentioned.  We  have  so  far  dealt  with 
three  stable  triple  points,  viz.  A,  B  and  C,  and  with  five  stable 
equilibrium  lines  AB,  BE,  AC,  CF,  CB.  There  is  yet  another 
stable  equilibrium  line,  namely  AD.  If  we  start  again  at  A 
with  the  three  phases,  vapour,  monoclinic  and  rhombic,  and 
lower  the  temperature,  it  will  be  found  that  the  monoclinic 
is  transformed  completely  into  rhombic,  and  the  rhombic  and 
vapour  can  be  kept  in  equilibrium  at  a  series  of  lower  tem- 
peratures by  suitably  lowering  the  pressure  (i.e.  an  automatic 
effect  of  the  system  itself  by  the  lowering  of  its  own  vapour 


ONE-COMPONENT  SYSTEMS  269 

pressure).  In  this  way  we  pass  along  AD.  Besides  these 
transformations  of  stable  forms,  it  is  also  possible  to  observe 
unstable  equilibrium  points.  Thus  the  point  H  (which  can 
be  reached  by  rapidly  heating  rhombic  sulphur  until  it  melts) 
is  the  unstable  triple  point  corresponding  to  the  three  phases, 
rhombic,  liquid,  and  vapour.  The  point  H  is  obtained  by 
producing  the  stable  rhombic — vapour  line  AD,  the  stable  liquid 
—vapour  line  CF,  and  the  stable  rhombic— liquid  line  BE.  In 
heating  rhombic  quickly  in  presence  of  its  vapour  we  start 
along  DA,  and  instead  of  rhombic  changing  into  the  stable 
monoclinic  variety  at  A,  the  system  rhombic — -vapour  can  be 
carried  along  the  line  AH  to  H,  at  which  point  liquid  makes 
its  appearance.  Similarly,  if  we  had  started  somewhere  on  the 
line  BE  with  rhombic  and  liquid  in  equilibrium,  and  lowered 
the  temperature  and  the  pressure  correspondingly  we  would 
reach  B,  at  which  point  monoclinic  should  appear.  If  the 
temperature  and  pressure  changes  are  rapid,  however,  we  can 
carry  the  system  rhombic — liquid  down  to  much  lower  tempera- 
tures and  pressures,  namely,  along  BH  to  H.  In  a  similar 
manner,  we  can  super-cool  the  liquid  at  C  so  that  the  liquid 
vapour  system,  which  has  been  passing  along  FC  can  be  made 
to  pass  as  far  as  H.  That  H  represents  an  unstable  state  is 
shown  by  the  fact  that  if  we  reach  H,  by  any  of  the  routes 
mentioned,  and  keep  the  system  (rhombic — liquid — vapour)  in 
contact  for  a  sufficient  time,  the  system  will  completely  trans- 
form itself  into  monoclinic  and  vapour,  the  path  followed  being 
a  vertical  line  down  until  the  AC  curve  is  reached  at  the 
temperature  of  H,  and  the  vapour  pressure  corresponding  to 
this  temperature.  As  regards  the  stability  of  monoclinic 
sulphur,  it  will  be  seen  from  the  figure  that  it  is  limited  on  all 
sides.  That  is,  it  can  only  exist  as  a  stable  phase  within  certain 
fixed  temperature  and  pressure  limits.  The  line  GA  represents 
the  unstable  equilibrium  curve  of  monoclinic — vapour  system. 
The  system  ought,  on  being  cooled  at  A,  to  change  into  rhombic 
and  follow  the  line  AD.  That  AG  is  unstable  is  shown  by  the 
fact  that  on  keeping  monoclinic  in  contact  with  vapour  at  a 
temperature  corresponding  to  some  point  on  AG  the  system 
will  entirely  transform  itself  into  rhombic  and  vapour.  Its 


270       A   SYSTEM  OF  PHYSICAL    CHEMISTRY 

path  will  be  represent^  by  a  vertical  line  downwards  (at  con- 
stant temperature)  until  the  vapour  pressure  takes  on  the  value 
corresponding  to  that  of  the  curve  AD  at  the  same  tempera- 
ture. In  the  cases  of  the  majority  of  substances  it  will  be 
observed  that  melting  of  the  solid  eventually  occurred  on 
raising  the  temperature  sufficiently,  and  this  liquid  could  be 
entirely  vaporised  on  further  raising  the  temperature.  It 
may  happen,  however,  that  the  complete  transition  from  solid 
to  gas  (i.e.  sublimation)  on  raising  the  temperature,  takes  place 
without  the  appearance  of  the  liquid  phase.  If  the  sublima- 
tion pressure  of  a  substance  is  greater  than  the  atmospheric 
pressure  at  any  temperature  below  the  point  of  fusion,  then  the 
substance  will  sublime  without  melting  when  heated  in  an 
open  vessel,  and  fusion  will  only  be  possible  at  a  pressure 
higher  than  atmospheric.  Red  phosphorus  and  arsenic  are 
instances  of  this  behaviour. 

The  component  sulphur  is  thus  capable  of  giving  rise  to 
the  following  systems  : — 


Invariant  systems :   Triple  points. 

{A  corresponding  to  rhombic — monoclinic — vapour 
B  „  rhombic — monoclinic — liquid 

C  „  monoclinic — liquid — vapour 

Unstable     H  „  rhombic — liquid — vapour 

Univariant  systems :  Curves. 

(AD  corresponding  to  rhombic — vapour 

I  AB  „  rhombic — monoclinic 

AC  „  monoclinic — vapour 

CF  „  liquid — vapour 

CB  „  monoclinic — liquid 

BE  ,,  rhombic — liquid 

AG  ,,  monoclinic — vapour 

AH  ,,  rhombic — vapour 

HC  „  liquid — vapour 

BH  „  rhombic — liquid 


Unstable 


Stable 


ONE-COMPONENT  SYSTEMS  271 

Bivariant  systems :  Areas. 

Space  to  the  left  of  DABE — rhombic 

„         „      right  of  FCBE — liquid 
Space  below  DACF— vapour 
Area  ABC — monclinic 


It  will  be  observed  that  the  Phase  Rule  predicts  that  four 
phases,  such  as  rhombic,  monoclinic,  liquid,  vapour,  cannot 
possibly  coexist  in  equilibrium  since  there  is  only  one  com- 
ponent, and  this  would  make  /  negative.  As  regards  the  slope 
of  the  lines  such  as  AB,  AC,  BC,  the  principle  of  Le  Chatelier- 
Braun  will  allow  us  to  predict  the  general  trend  if  we  know 
beforehand  the  densities  or  specific  volumes  of  the  phases. 
Thus  the  curve  AB  (monoclinic — rhombic)  slopes  to  the  right 
on  increasing  the  pressure.  If  we  start  with  any  point  on  AB 
and  increase  the  pressure,  keeping  the  temperature  constant,  it 
will  be  found  that  the  monoclinic  will  change  into  rhombic, 
i.e.  we  will  pass  along  a  vertical  line,  which  takes  us  right  into 
the  field  of  rhombic.  This  must  mean  that  the  specific  volume 
of  rhombic  is  less  than  that  of  monoclinic.  A  similar  relation 
holds  for  the  transformation  of  monoclinic  into  liquid  (BC), 
the  slope  of  the  curve  showing  that  the  monoclinic  has  the 
smaller  specific  volume  than  the  liquid,  i.e.  the  reverse  of  the 
case  ordinary  ice  I.  and  water. 

ONE-COMPONENT  SYSTEMS  (contitmed). — THE  COMPONENT  TIN. 

Tin  is  known  to  exist  in  two  solid  forms,  "  white "  and 
"  grey,"  as  well  as  in  the  liquid  and  vapour.  We  thus  have  a 
case  analogous  to  that  of  sulphur.  The  interesting  point  is 
the  determination  of  the  transition  point,  white  to  grey  (i.e.  the 
triple  point,  white — grey— vapour),  which  can  be  effected  in 
this  case  by  an  electrical  method.  If  we  set  up  a  cell  contain- 
ing an  aqueous  solution  of  tin  salt,  one  electrode  being  white 
tin,  the  other  grey  tin,  in  general  an  e.m.f.  will  be  given  by 
the  cell.  This  is  due  to  the  fact  that  the  electrodes  are  not 
electromotively  identical,  i.e.  they  do  not  possess  the  same  solu- 
tion pressures.  The  two  electrodes  not  being  identical,  there 


272       A    SYSTEM   OF  PHYSICAL   CHEMISTRY 

is  a  tendency  for  a  chemical  reaction  of  some  sort  to  occur, 
in  this  case  the  transformation  of  white  tin  into  grey  tin,  or 
vice  versti.  The  fact  that  substances  are  not  in  equilibrium 
with  one  another  is  always  manifested  by  the  existence  of  an 
e.m.f.,  if  the  nature  of  the  system  allows  a  cell  to  be  set  up. 
This  is  dealt  with  at  greater  length  in  the  chapter  on 
"Affinity."  If,  however,  white  and  grey  tin  were  in  equili- 
brium, no  e.m.f.  would  be  produced.  At  the  transition  point 
this  equilibrium  is  obtained,  and  hence  the  transition  point 
will  be  that  temperature  at  which  the  cell  gives  no  e.m.f.  By 
placing  the  cell  in  a  bath,  the  temperature  of  which  can  be 
altered,  the  transition  point  can  then  be  accurately  determined. 
Cohen  and  van  Eyk  found  it  to  be  +  20°  C.  Below  this 
temperature  grey  tin  is  stable.  The  investigation  of  the  trans- 
formations of  tin,  and  the  causes  of  the  phenomenon  known  as 
"tin  pest,"  has  been  made  in  particular  by  Cohen  and  his 
collaborators  (cf.  Zeitsch.  physik.  C7iem.,  30,  601,  1899, 
and  several  papers  in  recent  years). 

Besides  the  transition  of  grey  tin  into  white  tin,  the  com- 
ponent tin  shows  another  transition  at  a  very  much  higher 
temperature.  Ordinary  white  tin  is  tetragonal,  but  at  a 
temperature  of  about  203°  C.,  A.  Smits  and  H.  1.  de  Leeuw 
(Proc.  k.  Akad.  Wetensch,,  Amsterdam,  1912,  15,  676)  have 
shown  that  there  is  a  transformation  into  a  brittle  form, 
crystallising  in  the  rhombic  system.  It  is  difficult  to  bring 
this  change  about,  but  it  can  be  catalysed  by  the  addition  of 
small  quantities  of  mercury.  This,  however,  has  the  serious 
disadvantage  of  depressing  the  transition  temperature.  This 
transition  was  observed  by  dilatometric  measurements,  not  by 
electromotive  force. 

TWO-COMPONENT  SYSTEMS. 

Before  taking  up  the  behaviour  of  any  particular  system  in 
detail  it  is  necessary  to  consider  some  of  the  phenomena 
associated  with  the  physical  properties  of  compounds  and 
mixtures  (liquid  and  solid  solutions),  and  to  see  how  these 
properties  (such  as  fusion  and  solidification  temperatures,  boil- 
ing points,  vaporisation,  and  solubility)  are  able  to  afford  us 


TWO-COMPONENT  SYSTEMS  273 

valuable  information  regarding  the  chemical  changes  which  a 
system  may  undergo. 

Thus  consider  the  two-component  system  H2O — NaCl, 
the  special  case  being  a  dilute  solution  of  salt  in  water  at  a 
given  temperature.  Let  us  suppose  the  temperature  of  the 
solution  is  lowered.  At  a  given  temperature  ice  (solid  H2O 
ice  I.)  will  make  its  appearance.  This  is  the  so-called  freezing 
point  of  the  solution.  It  will  be  observed,  however,  that  this 
is  not  a  "  sharp  freezing  or  melting  point,"  i.e.  the  solution  as 
a  whole  cannot  be  made  to  solidify  at  the  temperature  at 
which  ice  first  made  its  appearance.  It  is  necessary  to  go  on 
lowering  the  temperature,  more  and  more  ice  being  therefore 
precipitated.  Analysis  will  show  that  in  this  case  (as  in  the 
majority  of  inorganic  salts  and  water)  the  solid  which  comes 
out  is  pure  ice.  There  is  no  salt  present  in  the  solid  phase. 
As  the  solid  phase  increases  in  bulk  the  solution  naturally 
increases  in  concentration  (in  respect  of  the  salt)  and  we 
already  know  that  a  concentrated  solution  freezes  at  a  lower 
temperature  than  a  dilute  one.  This  change  in  the  concen- 
tration of  the  salt  is  the  cause  of  the  progressive  lowering  of 
freezing  point,  which  any  solution  of  this  nature  will  show. 
This  lowering  will  not,  however,  go  on  without  limit.  A 
temperature  is  finally  reached,  at  which  the  salt  has  reached  its 
limit  of  solubility  in  the  liquid,  and  both  solid  salt  and  ice  are 
precipitated,  in  the  same  proportion  as  they  existed  in  the 
solution.  At  this  temperature  the  whole  system  will  go  solid. 
This  therefore  represents  the  lowest  possible  value  for  the 
"  freezing  point "  of  the  solution.  It  is  called  the  cryohydric 
or  eutectic  point  or  temperature.  Microscopic  analysis  will 
show  that  the  cryohydric  solid  is  in  this  case  heterogeneous •,  i.e. 
it  consists  of  ice  crystals  lying  side  by  side  with  salt  crystals, 
but  there  is  no  solid  solution  formed.  The  salt  crystals  are, 
however,  in  this  case  not  anhydrous  salt,  but  salt  in  the  form 
of  the  dihydrate,  NaCl .  2H2O.  The  behaviour  of  the  system 
is  best  shown  by  the  aid  of  the  diagram  (Fig.  76).  It  will  be 
seen  that  the  cryohydric  point  C  is  simply  the  point  at  which 
the  solubility  curve  of  the  salt  DC  cuts  the  freezing  point 
curve  of  the  solution.  The  existence  of  such  cryohydric 

T.P.C. — ii.  T 


274       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

points  is  of  immense,  importance  in  connection  with  the 
efficiency  of  freezing  mixtures.  No  freezing  mixture  can  cool 
itself  down  below  the  cryohydric  point  of  the  particular  solvent 
and  solute  employed.  [This  automatic  cooling  effect  is,  of 
course,  of  different  origin  to  the  external  cooling  we  have 
supposed  in  the  above.  In  freezing  mixtures,  say  of  ice  and 
common  salt,  the  salt  dissolves  in  some  of  the  liquid  (moisture) 
which  always  adheres  to  the  ice  crystals.  This  solution  and 
the  ice  are  not,  however,  in  equilibrium,  say,  at  o°  C.  There 
would  only  be  equilibrium  if  the  temperature  were  lower. 


-30 


— »-  Concentration,  of  Salt 
in,  solution, 

FIG.  76. 

Some  of  the  ice  therefore  melts,  and  to  do  this  the  latent  heat 
of  fusion  has  to  be  supplied  by  the  system  itself,  which  thereby 
cools.  Further  salt  now  dissolves  and  the  same  process  is 
repeated,  with  the  result  that  the  temperature  falls  still  further. 
The  temperature  can  never  fall,  however,  below  the  cryohydric 
point,  since  at  this  temperature  the  solution  being  saturated  the 
salt  has  no  further  tendency  to  dissolve.]  The  close  analogy 
between  "  freezing  point "  curves  and  t{  solubility  curves  "  will 
be  clear  from  the  figure.  In  the  case  which  we  have  considered 
we  started  with  a  solution,  the  composition  of  which  (in  respect 
of  salt)  lay  to  the  left  of  C,  and  at  room  temperature,  say.  If, 
on  the  other  hand,  we  had  started  with  a  concentrated  solution 


TWO-COMPONENT  SYSTEMS  275 

of  salt  (to  the  right  of  C),  and  at  room  temperature,  and  had 
cooled  this  down,  at  a  certain  temperature  solid  salt,  either 
anhydrous  or  in  the  dihydrate  form,  would  have  precipitated 
itself.  This  can  be  equally  regarded  as  the  freezing  point  of 
the  solution,  or  the  solubility  point  (temperature),  though  it  is 
more  usual  to  regard  it  as  the  latter.  In  this  case  also,  the 
solution  would  not  become  entirely  solid  at  the  temperature  at 
which  salt  first  appeared,  but  as  the  temperature  was  lowered 
more  and  more  salt  would  be  precipitated  until  the  tempera- 
ture C  was  again  reached,  and  ice  as  well  as  salt  is  simul- 
taneously precipitated.  Actual  analysis  of  the  solid  phase 
which  separates  out  from  concentrated  solutions  of  salt  has 
shown  that  above  the  temperature  -f-°'I5°C.  the  solid  is 
anhydrous  NaCl,  below  this  temperature  the  solid  is  the  com- 
pound NaCl .  2H2O.  The  temperature  0*15°  C  is  therefore 
the  transition  point  of  the  dihydrate  into  the  anhydrous  form. 
Above  this  temperature  we  say  that  the  anhydrous  salt  is  the 
stable  form,  below  this  temperature  the  dihydrate  is  stable. 
It  will  be  observed  that  the  relative  slope  of  the  two  curves, 
CD  and  DB,  is  in  agreement  with  the  generalization  which  we 
have  reached  on  thermodynamical  grounds,  viz.  that  at  a  given 
temperature  the  unstable  form  has  a  greater  solubility  than  the 
stable  form.  Thus,  suppose  the  line  CD  produced  upwards 
into  the  region  where  the  dihydrate  is  unstable.  It  will  be 
seen,  on  selecting  any  temperature  and  drawing  vertical  lines 
downward  from  each  curve  to  the  concentration  axis,  that  at 
one  and  the  same  temperature  the  solubility  of  the  dihydrate 
(if  it  could  be  got  into  this  position  at  all)  is  greater  than  that 
of  the  anhydrous  salt.  Similarly  at  temperatures  below  D,  the 
anhydrous  salt  (which  is  now  unstable)  has  a  greater  solubility 
than  the  dihydrate. 

Formation  of  Solid  Solutions. 

In  the  above  case  the  two  constituents  are  precipitated 
separately  (at  the  cryohydric  point  together,  but  hetero- 
geneously).  In  some  cases  both  constituents  are  precipitated 
simultaneously  (at  all  "  freezing  points"),  and  the  solid  is  a 


276       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

homogeneous  solid  solution.  The  formation  of  solid  solutions, 
in  which  the  components  are  mutually  miscible  in  all  propor- 
tions, can  give  rise  to  three  types  of  curves,  the  first  two  of 
which  are  important.  In  the  first  type  the  freezing  or  melting 
points  of  the  mixture  lie  between  the  freezing  points  of  the 
two  pure  constituents.  This  is  illustrated  in  the  accompanying 
diagram  (Fig.  77),  which  shows  the  behaviour  of  mixtures  of 
chloro-cinnamic-aldehyde  and  bromo-cinnamic-aldehyde,  or  by 
alloys  of  gold  and  platinum.  The  freezing  point  of  the  pure 
chloro-compound  is  31*22°  C.,  that  of  the  bromo-compound  is 


69-5611 


31-22 


IOO°7oCh/oro- 
Comp<* 


Composition  of 


IOO%  Bromo- 


Solid  Solu.tion  ("mixed  crystals") 
FIG.  77. 

69*5 6°  C.  Suppose  a  little  chloro-compound  is  added  to  the 
pure  bromo-compound.  The  freezing  point  is  lowered.  In 
those  cases  in  which  the  "solute"  crystallises  out  with  the 
solvent,  the  lowering  of  freezing  point  —  AT  is  given  by  the 
expression — 


where  x1  is  the  concentration  of  solute  in  the  liquid  phase,  and 
x%  the  concentration  in  the  solid.  In  order  that  a  lowering  of 
freezing  point  may  take  place  it  is  necessary  that  x-^>  x^  i;t- 
that  the  concentration  of  the  chloro-compound  shall  be  greater 
in  the  liquid  solution  than  it  is  in  the  solid.  To  represent  the 


TWO-COMPONENT  SYSTEMS  277 

behaviour  of  the  system  consisting  of  liquid  and  solid  solu- 
tions, it  is  thus  necessary  to  have  TWO  curves,  which  in  certain 
cases  may^  indeed \  lie  very  close  together  but  can  never  be  identical.^- 
One  curve  gives  the  composition  of  the  liquid  solution,  and  is 
known  as  the  liquidus  curve  ;  the  other  gives  the  composition 
of  the  corresponding  solid  solutions,  and  is  known  as  the 
solidus  curve.  In  the  above  case  the  upper  curve  gives  the 
composition  of  the  liquid  solution,  the  lower  that  of  the  solid 
solution.  The  impossibility  of  both  curves  being  identical, 
except  at  the  limits,  is  evident  from  a  consideration  of  the 
above  formula,  for  identity  of  the  two  curves  would  mean 
identity  in  composition  of  the  two  phases,  and  this  would 
mean  that  x±  =  x2  and  therefore  —  AT  =  o.  That  is,  the 
addition  of  the  second  component  would  have  no  effect  on 
the  freezing  point,  which  is  evidently  quite  unthinkable  since 
the  pure  substances  have  quite  different  melting  points,  and 
we  must  be  able  to  pass  from  one  extreme  to  the  other.  The 
problem  of  the  relative  position  of  the  solidus  and  liquidus 
curves  is  solved  by  means  of  the  thermodynamical  equation 
given.  Stated  in  general  terms  the  conclusion  is  as  follows  : — 

"  At  any  given  temperature  the  concentration  of  that  component \ 
by  the  addition  of  which  the  freezing  point  is  depressed  is  greater 
in  the  liquid  than  in  the  solid  phase  ;  or  conversely ',  the  concentra- 
tion of  that  component  by  the  addition  of  which  the  freezing  point 
is  raised  is  greater  in  the  solid  than  in  the  liquid  phase"  The 
comparison  of  the  concentration  of  the  two  phases  must,  of 
course,  be  made  at  the  same  temperature.  This  is  represented, 
for  different  temperatures,  at  different  parts  of  the  curve,  by 
the  lines  -Is,  /Y,  and  /"/',  etc. 

It  will  be  evident  that  in  a  case  such  as  that  just  con- 
sidered the  freezing  point  of  the  solution  at  any  composition 
(except  the  two  extreme  points)  will  not  be  sharp^  i.e.  the 
system  as  a  whole  cannot  be  solidified  at  one  temperature, 
there  being  instead  a  progressive  change  of  freezing  point.'2 

1  Except  in  the  special  case  in  which  the  melting  points  of  both   con- 
stituents are  identical,  and  then  the  line  is  horizontal  on  a  To;  diagram 
similar  to  that  given. 

2  Ordinary   solutions,    such    as    salt    and    water,   exhibit    the    same 


278        A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

This  is  due  to  the  fact  that  at  no  point  is  the  composition  of 
solid  and  liquid  identical,  except  of  course,  in  the  case  of  each 
pure  substance,  at  the  ends  of  the  curve.  Referring  to  the 
preceding  diagram,  suppose  that  the  system  under  examination 
is  a  liquid  solution  of  the  two  components  at  the  composition 
and  temperature  denoted  by  the  point  x.  Suppose  the  tem- 
perature is  lowered,  the  system  begins  to  pass  along  the  vertical 
line  xx' .  On  reaching  the  point  /',  some  solid  is  deposited, 
the  solid  having  the  composition  s'.  On  lowering  the  tem- 
perature further,  solid  comes  out  of  solution,  the  composition 
of  the  solid  phase  passing  along  //',  the  composition  of  the 
liquid  being  simultaneously  represented  by  a  series  of  points 
on  the  line  I'l" .  If  the  temperature  be  still  further  lowered, 
the  system  as  a  whole  may  go  solid.  There  is  thus  a  greater 
or  smaller  temperature  interval,  I's"  "the  crystallisation 
interval,"  which  must  be  passed  through  between  the  first  appear- 
ance of  solid  and  the  final  complete  solidification.  It  should 
be  noted  that  in  order  to  have  the  solid  phase  homogeneous 
as  regards  concentration  of  the  components  it  is  necessary  to 
allow  the  operation  of  cooling  to  go  on  very  slowly,  for  as  the 
composition  of  the  freshly  deposited  solid  is  continuously 
changing  (along  s's")  there  must  be  diffusion  in  the  solid  itself 
to  give  an  average  composition.  To  bring  this  diffusion  about 
it  is  usual  to  heat  the  solid  to  a  temperature  somewhat  below 
its  melting  point.  This  is  known  in  alloy  work  as  "  anneal- 
ing "  (cf.  Desch,  Metallography ',  pp.  46  seq.).  Fractional 
crystallisation  of  mixed  crystals  (as  solid  solutions  are  some- 
times termed)  can  be  effected  by  separating  off  the  solid  mass 
which  appears  at  /,  say,  remelting  this,  and  allowing  it  to 
deposit  other  crystals  represented  by  /",  separating  these  in 
turn  and  again  melting  and  recrystallising.  In  these  processes 
the  solid  is  gradually  becoming  richer  in  the  bromo-com- 
pound  (taking  the  particular  case  studied).  Theoretically, 
however,  the  separation  can  never  be  complete. 

The  second  type  of  behaviour  exhibited  by  solid  solutions 

phenomenon  (for  the  same  reason).  The  "freezing  point"  of  such  a 
solution  is  therefore  the  temperature  at  which  a  very  small  quantity  of  some 
constituent  begins  to  come  out  in  the  solid  form. 


TWO-COMPONENT  SYSTEMS 


279 


ay  be  illustrated  by  the  system  mercuric  iodide  and  mercuric 
bromide,  which  are  also  capable  of  forming  solid  solutions 
with  one  another  in  all  proportions.  The  behaviour  of  this 
system  is,  however,  different  from  that  already  considered, 
though  here  also,  as  we  shall  see,  there  is  no  compound  formed. 
Mercuric  iodide  melts  at  255°  C.,  mercuric  bromide  at  236'5°  C. 
The  melting  point  of  mixtures  of  these  two,  however,  do  not 
lie  on  lines  connecting  these  two  temperatures.  Instead  the 
behaviour  is  shown  in  the  diagram  (Fig.  78),  which  is  a 
temperature  concentration  diagram.  Analysis  shows  that  the 


255° 


24-5° 


235' 


225 


215 


Temperature  -  Composition  Diagram 
Mercuric  Iodide  &  Bromide.  Mixed 
crystals  but  no  compound  formed. 

FIG.  78. 

composition  of  the  solid  and  liquid  forms  in  equilibrium  with 
one  another  (at  a  given  freezing  point)  are  not  the  same, 
except  at  one  temperature,  namely,  C,  the  lowest  temperature  at 
which  the  system  formed  from  these  two  components  can  be 
made  to  melt  or  to  freeze.  The  composition  of  both  solid  (s) 
and  liquid  (/)  are  shown.  Suppose  we  have  chosen  a  mixture 
of  75  per  cent,  mercuric  iodide,  and  25  per  cent,  mercuric 
bromide,  and  have  heated  it  until  it  is  completely  liquid. 
Now  begin  lowering  the  temperature,  thereby  allowing  some 
to  resolidify,  and  analyse  both  phases.  It  will  be  found  that 
the  liquid  has  the  composition  denoted  by  /  and  the  solid 


28o        A   SYSTEM  OF  PHYSICAL   CHEMISTRY 


(which  is  a  homogeneous  solid  solution)  has  the  composition 
denoted  by  s.  It  is  impossible  to  cause  the  whole  liquid  to 
solidify  at  this  temperature,  for  the  solid  being  richer  in  iodide 
than  the  solution  the  process  of  solidification  entails  the 
gradual  impoverishment  of  the  solution  as  regards  HgI2,  and 
the  freezing  point  falls  until  we  reach  s".  We  here  regard  the 
HgI2  as  the  "  solvent,"  the  HgBr2  as  the  solute  (because  there 
happens  to  be  much  more  HgI2  present),  though  of  course  the 
terms  are  always  interchangeable.  At  the  point  C  the  solid 
separating  is  identical  in  composition  with  the  solution,  and 
the  whole  system  can  solidify  sharply.  This  only  differs 
from  the  cryohydric  point  in  the  ice-salt  case,  in  the  fact  that 
in  the  present  system  the  point  C  is  a  point  on  a  continuous 
curve,  and  the  solid  is  a  homogeneous  solution  of  both  com- 
ponents. In  the  above  cases  there  is  no  evidence  of  the 
formation  of  a  true  compound.  Further,  in  these  cases  no 
melting  point  was  observed  higher  than  that  of  either  pure 
constituent ;  nor,  indeed,  was  a  curve  obtained  having  a  por- 
tion concave  (towards  the  concentration  axis).  This  type 

of  curve  is  obtained  in  the 
case  of  solid  solutions  of 
optical  isomerides,  namely, 
d  and  /  carvoxime,  in  which 
no  compound  is  formed 
although  the  curve  passes 
through  a  maximum,  i.e.  has 
a  melting  point  higher  than 
that  of  either  single  com- 
ponent. This  curve  is  shown 
in  the  figure  (Fig.  79).  The 
composition  of  the  solid  and 
liquid  phases  are  different 
except  at  the  maximum  point. 
It  will  be  observed  that  in 

this  case  there  are  no  cutectics  to  either  side  of  the  maximum. 
Desch  states  that  in  alloys  this  type  of  curve  is  never  found. 
The  significance  of  the  absence  of  eutectic  points  on  either 
side  of  the  maximum  will  be  clear  from  the  consideration  in 


100% 

A        Composition  of 
Solid  Solutions 


B 


TWO-COMPONENT  SYSTEMS 


281 


the  next  paragraph  of  the  system  a-naphthylamine  and  phenol 
which  will  now  be  discussed. 


Formation  of  Compounds. 

In  the  case  of  a-naphthylamine  and  phenol  no  solid 
solution  is  formed,  but  instead  a  true  compound  of  the  two 
components  (J.  C.  Philip,  Trans.  Chem.  Soc.>  83,  821,  1903), 
cf.  Fig.  80.  Phenol  melts  at  40-4°  C.,  and  a-naphthylamine  at 
48*3°  C.  On  adding  some  of  the  amine  to  liquid  phenol,  the 
freezing-point  of  the  solution  is  lowered,  and  we  can  then  pass 


IOO%A  (Phenol) 

50 


30 


20 


10 


100%  B|£r  Napthy/amineJ 

B 


20 


50 

FIG.  80. 


80 


along  AC  by  adding  successive  amounts  of  the  amine.  The 
solid  which  separates  is  pure  phenol  until  the  point  C  is 
reached,  at  which  the  whole  system  may  be  caused  to  solidify. 
This  must  be  a  eutectic  point,  some  other  solid  besides  the 
phenol  having  evidently  separated  out.  Analysis  shows  the 
presence  of  both  phenol  and  the  amine,  but  this  leaves  us  in 
doubt  as  to  what  the  solid  is.  The  case  may  only  be 
analogous  to  that  of  ice  and  salt.  On  further  addition  of  the 
amine  the  temperature  of  equilibrium  between  solid  and  liquid 
phases  rises.  At  D  the  curve  passes  through  a  maximum 
(28-8°  C.),  and  on  further  addition  of  amine  again  falls. 
Analysis  shows  that  at  D  the  composition  of  the  solid  and 


282        A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

liquid  are  identical  {ias  was  also  the  case  at  C).  Further,  at 
D  this  composition  of  both  liquid  and  solid  is  exactly  given 
by  the  formula  C6H5OH,  C10H7NH2.  By  adding  amine  we 
finally  reach  another  point  E,  at  which  the  system  can  be 
solidified  entirely  without  change  of  temperature.  Starting 
with  pure  amine  and  adding  phenol,  we  can  obtain  a  series  of 
freezing  points  along  BE.  E  is  the  second  eutectic,  the  pro- 
longation of  the  lines  AC  and  BE,  below  C  and  E  respectively, 
denote  metastable  conditions  of  the  system,  which  in  this  case 
can  be  reached  owing  to  the  slow  rate  of  transformation  from 
one  form  to  another.  Now  we  know  that  the  addition  of  any 
foreign  substance  will  lower  the  freezing-point  of  a  pure 
substance.  This  is  what  has  happened  at  D.  In  fact,  if  we 
assume  the  existence  of  a  compound  consisting  of  phenol  and 
a-naphthylamine  in  equi-molecular  proportions,  the  shape  of 
the  curve  ACDEB  is  what  we  would  expect.  D  is  the 
melting-point  of  the  compound,  and  it  can  be  lowered  either 
by  the  addition  of  phenol  or  of  a-naphthylamine.  In  this  way 
we  can  account  for  the  fall  on  both  sides  to  C  and  E  respec- 
tively. The  solids  present  at  the  eutectic  at  C  are  phenol 
and  C6H5OH,  C10H7NH2.  The  solids  present  at  the  eutectic 
at  E  are  the  same  compound  and  the  amine.  The  concavity 
of  the  curve  CDE  with  regard  to  the  concentration  axis,  the 
curve  ending  in  two  eutectics,  is  very  characteristic  of  the 
presence  of  a  compound.  The  same  behaviour  is  shown  by 
several  alloys,  e.g.  magnesium  and  tin,  the  compound  Mg2Sn 
being  produced,  cf.  Desch's  Metallography.  (The  solution  or 
liquid  in  contact  with  the  pure  solid  at  D  may  be  regarded 
simply  as  the  molten  solid  itself.)  This  is  a  good  illustration 
of  how  purely  physical  methods  can  be  employed  to  detect 
chemical  combination.  As  we  observe,  there  is  a  good 
theoretical  reason  for  considering  that  a  concave  curve  of  the 
above  type  may  be  taken  as  evidence  of  the  formation  of  a 
compound  between  the  components.  A  convex  curve  show- 
ing a  minimum  melting-point,  on  the  other  hand  (as  in  the 
system  HgI2,  HgBr2),  does  not  mean  the  formation  of  a  com- 
pound, though  it  does  indicate  the  formation  of  a  homo- 
geneous eutectic.  If  the  minimum  point  had  really  been  a 


TWO-COMPONENT  SYSTEMS  283 

point  of  intersection  of  two  separate  lines  (a  freezing-point 
line  and  a  solubility  line,  as  in  the  H2O— Nad  case)  we 
should  be  justified  in  considering  the  system  at  this  point  as 
a  heterogeneous  eutectic. 

Liquid  Mixtures. 

So  far  some  instances  have  been  given  of  the  behaviour 
of  different  sorts  of  two-component  systems  as  manifested  by 
the  phenomena  of  freezing  point  and  solubility.  One  or  two 
instances  of  the  behaviour  exhibited  by  two-component  systems 
in  the  change  from  the  liquid  to  the  vapour  state  may  be 
briefly  indicated.  The  most  interesting  problem  is  that  of  the 
distillation  of  mixtures.  When  the  two  components  are  quite 
immiscible  the  total  vapour  pressure  =  the  sum  of  the  pres- 
sures of  the  two  substances  (measured  separately).  In  the 
process  of  distillation  they  are  quite  without  influence  on  each 
other.  When  two  liquids  are  partially  miscible  a  distillate  of 
definite  composition  at  a  constant  temperature  is  obtained  as 
long  as  the  two  layers  are  present  in  the  distillation  flask. 
When  one  layer  only  remains  in  the  flask,  and  distillation  is 
continued,  the  distillation  proceeds  as  in  the  case  to  be  con- 
sidered next,  namely,  that  of  two  liquids  completely  miscible 
in  all  proportions. 

The  behaviour  of  systems  of  this  type  varies  according  to 
the  chemical  nature  of  the  components.  The  presence  of  each 
constituent  lowers  mutually  the  vapour  pressure  of  each  (i.e. 
the  usual  effect  of  the  presence  of  a  "  solute  "  upon  a  "  solvent "), 
and  the  boiling  point  may  be  higher  or  lower  than  the  boiling 
point  of  either  component,  depending  on  the  value  of  the  sum 
of  the  vapour  pressures,  for  under  ordinary  conditions  boiling 
sets  in  when  the  sum  of  the  vapour  pressures  =  i  atmosphere. 
In  general  the  composition  of  the  vapour,  and  therefore  of  the 
distillate  which  is  simply  condensed  vapour,  differs  from  that 
of  the  original  mixture.  In  a  simple  case  such  as  that  of 
methyl  alcohol  and  water,  the  water  being  in  considerable 
excess,  the  vapour  is  much  richer  in  the  alcohol  than  is  the 
liquid,  and  finally  by  distillation  pure  water  is  left  in  the 


284       A    SYSTEM   OF  PHYSICAL   CHEMISTRY 

retort.  In  certain  cases  we  meet  with  the  remarkable  phe- 
nomenon known  as  Mixtures  of  constant  boiling  point^  which 
can  be  distilled  at  constant  temperature  unchanged,  i.e.  the 
composition  of  the  vapour  or  distillate  remains  unchanged. 
Such  a  mixture  behaves  as  a  single  pure  substance,  and  was 
indeed  for  long  regarded  as  a  compound  of  the  two  com- 
ponents, until  it  was  shown  by  Roscoe  that  by  altering  the 
boiling  temperatures  (by  altering  the  external  pressure)  a 
series  of  constant  boiling  mixtures  could  again  be  obtained, 
each  of  which  differed,  however,  from  one  another  in  com- 
position. A  true  compound,  of  course,  could  not  alter  in 
composition  by  simply  altering  the  pressure.  A  constant 
boiling  mixture  is  one  which  possesses  either  a  greater  vapour 
pressure  than  that  of  any  other  mixture  of  the  two  components, 
or  possesses  a  smaller  vapour  pressure  than  that  of  any  other 
mixture.  Thus  96  per  cent,  ethyl  alcohol  +  4  Per  cent-  water 
possesses  a  lower  vapour  pressure,  and  therefore  a  higher 
boiling-point,  than  that  of  any  other  mixture  of  these  two 
components.  If  we  start  therefore  with  any  smaller  per- 
centage of  alcohol  than  this,  and  gradually  raise  the  tempera- 
ture distillation  will  occur  in  such  a  way  that  the  distilling 
liquid  loses  water  more  quickly  than  alcohol,  i.e.  the  distillate 
is  richer  in  water  than  the  original  mixture  until  the  com- 
position of  the  liquid  in  the  distilling  flask  reaches  the 
value  96  per  cent,  alcohol,  after  which  the  remainder  distils 
without  change  of  temperature  or  composition.  Similarly,  if 
we  started  with  a  mixture  of  98  per  cent,  alcohol,  the  process 
of  distillation  would  have  caused  excess  alcohol  (along  with 
some  water)  to  pass  over,  the  boiling-point  rising  until  the 
constant  boiling  mixture  was  again  reached.  The  mixture 
consisting  of  70  per  cent,  propyl  alcohol  and  30  per  cent, 
water  happens  to  possess  a  higher  vapour  pressure  than  any 
other  mixture  of  the  two,  so  that  if  we  start  distilling  a  mixture 
of  any  other  composition  the  distillate  which  comes  over  first, 
i.e.  that  which  possesses  the  lowest  boiling  point,  contains 
70  per  cent,  propyl  alcohol.  This  distils  at  a  constant  tem- 
perature until  one  or  other  of  the  constituents  entirely  dis- 
appears from  the  distilling  flask.  In  this  case  it  is  possible  to 


TWO-COMPONENT  SYSTEMS  285 

effect  a  complete  isolation  of  one  liquid  in  a  single  operation, 
which  was  impossible  in  the  case  of  ethyl  alcohol  and  water 
mixtures.  The  system  hydrochloric  acid  and  water  is  a  well- 
known  instance  of  such  constant  boiling  mixtures,  that 
containing  20*2  per  cent,  of  acid  having  a  minimum  vapour 
pressure  and  boiling  at  no°C.  under  atmospheric  pressure. 
This  behaviour  is  analogous  to  the  ethyl  alcohol — water  system. 
The  properties  of  liquid  mixtures  are,  however,  so  varied,  and 
of  so  specific  a  nature,  that  there  is  no  opportunity  to  do 
justice  to  this  subject  in  a  general  text-book.  The  reader 
is  therefore  referred  to  Sidney  Young's  Stoichiometry ,  in  this 
series,  and  likewise  to  the  same  author's  book,  Distillation ; 
also  Kuenen's  Verdampfung  und  Verfliissigung  der  Gemischen 
(in  Bredig's  Handbooks  of  Applied  Physical  Chemistry). 

Liquid  Crystals. 

In  dealing  with  transformation  from  one  state  to  another, 
an  important  case  still  remains  to  be  considered  although  its 
theoretical  significance  has  not  yet  been  generally  agreed  upon. 
The  phenomenon  referred  to  is  the  formation  of  so-called 
Liquid  Crystals.  As  early  as  1888  it  was  observed  by 
Reinitzer  that  in  the  case  of  two  solid  substances,  cholesteryl 
acetate  and  cholesteryl  benzoate,  each  possesses  the  property 
of  melting  sharply  at  definite  temperatures,  but  the  liquid 
instead  of  being  transparent  is  turbid  and  milky.  This  turbid 
liquid  has  a  definite  temperature  range  of  stability,  for  on 
heating  still  further,  the  system  becomes  quite  transparent  at 
a  given  definite  temperature.  The  same  phenomenon  is 
exhibited  by  a  few  other  substances,  notably  by  para-azoxy- 
anisole  and  para-azoxyphenetole.  The  phenomenon  is  a 
reversible  one  in  the  chemical  sense,  i.e.  on  cooling  the 
transparent  liquid  it  assumes  the  turbid  state  at  a  given 
temperature  and  this  in  turn  solidifies  to  the  solid,  also  at  a 
given  temperature.  The  turbid  system  is  liquid  in  the  sense 
that  it  flows  like  a  liquid  and  can  be  made  to  assume  the 
spherical  form  when  placed  in  a  liquid  of  the  same  density. 
It  also  exhibits  the  property  of  double  refraction  which  is 


286       A   SYSTEM  OF  PHYSICAL    CHEMISTRY 

characteristic  of  certain  crystals.  Hence  Lehmann  (who  has 
specially  devoted  himself  to  work  in  this  field)  coined  the 
name  "  liquid  "  crystals.  Discussion  has  centred  round  the 
problem  :  Is  the  turbid  state  heterogeneous  or  homogeneous  ? 
i.e.  is  it  one  phase  or  more  than  one  ?  Its  appearance  is  very 
similar  to  that  of  an  emulsion,  and  this  suggests  heterogeneity. 
On  the  other  hand  it  has  not  been  possible  to  effect  a  separa- 
tion of  one  phase  from  the  other,  and  further,  the  existence  of 
a  sharp  solidifying  point  and  a  sharp  clearing  point  are 
evidence  for  homogeneity.  A  definite  conclusion  has  not  yet 
been  reached. 


THE  TWO-COMPONENT  SYSTEM  :  SODIUM  SULPHATE— WATER. 

At  ordinary  temperatures  the  solid  which  crystallises  from 
a  saturated  aqueous  solution  of  sodium  sulphate  is  the  deka- 
hydrate  Na2SO4 .  ioH2O.  At  a  temperature  of  50°  C.  the 


20° 


4-0°         60° 

Terrup . 
FIG.  81. 


80°         IOO°C 


solid  which  separates  out  is  the  anhydrous  salt  Na2SO4.  It  is 
clear  that  there  must  be  a  transition  temperature  at  which  both 
deka-hydrate  and  anhydrous  salt  may  crystallise  out  together, 
since  both  are  in  equilibrium  at  the  transition  point.  Solu- 
bility measurements  have  indicated  this  point  very  clearly. 
The  results  obtained  are  shown  in  the  diagram  (Fig.  81). 
From  o°  C.  up  to  the  temperature  32-4°  C.  the  deka-hydrate  is 
the  stable  form,  its  solubility  being  indicated  by  the  ascending 
line.  From  32-4°  C.  onwards  the  anhydrous  salt  crystallises, 
and  this  salt  shows  the  phenomenon  of  retrograde  solubility, 


TWO-COMPONENT  SYSTEMS  287 

i.e.  the  solubility  diminishes  as  the  temperature  rises.  The 
point  E  (32*4°  C.)  is  the  transition  point.  This  is  a  good 
instance  of  how  solubility  measurements  can  be  utilised  to 
indicate  such  regions  of  stability.  In  addition  to  the  anhydrous 
salt  and  the  deka-hydrate,  another  solid  can,  however,  be 
prepared  under  certain  conditions,  namely,  the  hepta-hydrate 
Na2SO4.7H2O.  If  a  solution  of  sodium  sulphate,  saturated 
at  the  temperature  34°  approximately,  be  cooled  down  in  the 
absence  of  dust  particles  or  other  nuclei,  it  is  possible  to  reach 
the  temperature  17°  C.,  when  the  solid  which  crystallises  out 
will  be  found  to  be  the  hepta-hydrate.  Determinations  of  the 
composition  of  the  solutions  in  equilibrium  with  this  hydrate 
at  different  temperatures  from  the  region  o°  C.  allow  us  to 
plot  the  dotted  line  ascending  to  D.  The  solubility  curve  of 
the  hepta-hydrate  also  shows  a  sharp  change  in  direction  at  D, 
the  solubility  being  now  less  as  the  temperature  increases,  and 
the  solid  phase  separating  out  being  the  anhydrous  salt.  This 
is  indicated  by  the  dotted  line  DE,  which  continues  past  E  on 
without  a  break.  The  point  D  corresponds  to  temperature 
24*4°  C.,  P=  i8'9  mm.,  and  is  the  transition  temperature  of 
the  hepta-hydrate  into  the  anhydrous  salt.  At  D  two  solid 
phases  (the  anhydrous  salt  and  hepta-hydrate)  are  in  equi- 
librium with  saturated  solution  and  vapour.  Since  there  are 
four  phases,  and  the  system  is  a  two-component  one,  we  must 
have  f—o,  i.e.  the  system  is  invariant,  and  must  be  repre- 
sented by  a  point  on  the  diagram.  On  altering  one  of  the 
variables,  say,  temperature,  one  of  the  phases  must  disappear. 
The  phase  which  disappears  is  the  hepta-hydrate,  and  we 
find  the  system  travelling  along  the  line  DE.  The  transition 
point  D  and  the  lines  leading  to  it  are  dotted,  indicating 
meta-stability.  This  is  clear,  for  the  solubility  of  the  hepta- 
hydrate  is  greater  than  the  deka-hydrate  at  any  temperature, 
and  hence  the  hepta-hydrate  must  be  unstable  with  respect 
to  the  deka-hydrate.  Sodium  sulphate  forms  therefore  only 
one  stable  hydrate.  The  anhydrous  salt  is  for  the  same  reason 
unstable  with  respect  to  the  deka-hydrate  at  all  points  from  D 
to  E.  At  temperatures  higher  than  E,  however,  the  solu- 
bility of  the  deka-hydrate  would  be  greater  than  that  of  the 


288        A   SYSTEM  OF  PHYSICAL    CHEMISTRY 

anhydrous  salt,  i.e.  fr^om  E  onwards  the  anhydrous  salt  is  the 
stable  phase.  It  is  important  to  notice  that  although  we  speak 
of  the  temperature  32-4°  C.  (point  E)  as  the  transition  point 
from  deka-hydrate  to  anhydrous  salt,  this  limitation  to  the 
stability  has  reference  to  the  solid  form,  not  to  the  solutions. 
At  all  temperatures  (say  from  o°  to  50°)  the  aqueous  solution 
contains  in  all  probability  mixtures  of  anhydrous  molecules 
and  deka-hydrate  molecules,  there  being  an  equilibrium 
between  them  as  indicated  by  the  equation — 

Na2S04  +  ioH20  ^>  Na2S04  .  ioH2O. 

This  is  a  homogeneous  equilibrium  governed  by  the  mass- 
action  principle.  The  equilibrium  varies  with  the  temperature, 
but  there  is  no  reason  for  believing  that  in  the  solution  below 
32'4°,  deka-hydrate  molecules  alone  exist >  and  above  32*4° 
anhydrous  molecules  alone  exist.  Both  coexist  over  the 
temperature  range.  What  must  happen  is,  that  as  the  tem- 
perature rises  (say,  we  start  below  32-4°)  the  homogeneous 
equilibrium  shifts  over  to  the  left  so,  that  just  at  32*4°  the 
solution  is  saturated  with  respect  to  anhydrous  salt  as  well  as 
with  respect  to  deka-hydrate.  On  raising  the  temperature  by 
an  infinitesimal  amount  the  equation  likewise  shifts  more 
towards  the  left,  the  anhydrous  salt  being  now  super-saturated. 
Anhydrous  salt  crystallises  (assuming  retardation  is  prevented) 
and  at  the  same  time  by  the  principle  of  mass  action,  the  equi- 
librium tends  to  maintain  itself,  that  is  some  more  anhydrous 
salt  molecules  are  formed  in  the  solution  at  the  expense  of  the 
deka-hydrate  molecules,  these  being  in  turn  supplied  by  the 
solid  deka-hydrate.  The  solid  deka-hydrate  therefore  begins 
to  dissolve,  i.e.  disappear,  just  above  E.  The  solution  is, 
however,  again  super-saturated  with  respect  to  anhydrous  salt, 
and  this  further  precipitates  itself,  the  above  process  being 
repeated  until  all  the  solid  deka-hydrate  has  disappeared  and 
we  are  left  with  the  invariant  system  anhydrous  salt  solid — 
saturated  solution — vapour.  The  discovery  of  the  points  E 
and  D,  and  the  solubility  lines  leading  to  and  from  them, 
exhaust  by  no  means  our  information  respecting  the  behaviour 
of  this  two-component  system.  The  temperature-concentration 


TWO   COMPONENT  SYSTEMS 


289 


and  vapour  pressure-temperature  diagrams  have  recently  been 
extended  by  A.  Smits  and  J.  P.  Wuite  (Prof.  Roy.  Soc.  Am- 
sterdam^ 12,  244,  1909-1910).  The  complete  behaviour  of 
the  system  as  far  as  it  can  be  shown  on  a  temperature- 
concentration  diagram,  is 
illustrated  by  the  accom- 
panying figure  (Fig.  82),  in 
which  temperature  is  de- 
noted as  the  ordinates  and 
composition  (per  cent, 
sodium  sulphate)  by  the 
abscissae. 

If  we  commence  with 
pure  water  at  the  point  A 
(temperature  o°)  and  add 
sodium  sulphate,  we  can 
trace  out  the  line  AB,  which 
gives  the  temperatures  at 
which  ice  is  in  equilibrium 
with  (dilute)  solutions  of  the 
salt.  At  B  there  is  a  eutectic 
point,  the  solids  in  equi- 
librium being  ice  and 
Na2SO4.ioH2O.  If  the 
system  be  super-cooled,  no 
deka-hydrate  being  allowed 
to  form  at  B,  we  can  reach 
the  point  C  at  which  there  is 

another  eutectic  (a  metastable  one),  the  solids  in  metastable 
equilibrium  being  ice  and  Na2SO4  .  7H2O.  Returning  to  the 
point  B  and  raising  the  temperature  slightly,  the  ice  will  dis- 
appear, and  the  solid  phase  will  consist  of  dekahydrate.  By 
measurements  of  the  solubility  of  the  dekahydrate  at  various 
temperatures  we  pass  along  the  line  BE.  (The  same  pro- 
cedure at  C  would  have  taken  us  along  CD,  the  solid  being  the 
heptahydrate.)  At  E,  a  new  solid  makes  its  appearance,  viz., 
anhydrous  rhombic  Na2SO4.  E  is  therefore  another  eutectic 
(32-4°  C  and  30-8  mm.  Hg),  the  solids  in  equilibrium  being 
T.P.C. — n.  u 


FIG.  82. 


290       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

Na2SO4.ioH2O  and  Na2SO4  rhombic.  On  raising  the  tem- 
perature the  deka-hy3rate  disappears  and  we  follow  the  line 
ERF,  which  gives  the  solubility  of  rhombic  Na2SO4  as  a  func- 
tion of  temperature.  Starting  from  the  meta-stable  eutectic 
point  D  24*4°  C.,  we  can  similarly  follow  the  line  DERF.  It 
will  be  observed  that  the  solubility  of  Na2SO4  rhombic  at  first 
decreases  with  rising  temperature,  and  eventually  increases,  the 
minimum  being  about  125°  C.  Theoretical  reasoning  on  the 
significance  of  this  retrograde  curve  is  given  by  Smits,  /.£.,  12, 
227,  1909-10.  When  F  is  reached  a  sharp  change  in  direction 
is  experienced  by  the  solubility  curve,  this  being  due  to  the 
appearance  of  a  new  solid  phase  which  analysis  shows  to  be 
Na2SO4  monoclinic.  F  is  therefore  a  eutectic  point  (234°  C. 
27*5  atmospheres)  discovered  by  Nacken.1  On  passing  this 
temperature  the  monoclinic  is  the  stable  form  the  solubility 
of  which  decreases  as  the  temperature  rises.  Finally  we 
reach  the  point  P,  the  temperature  of  which  is  365°  C.  This 
is  the  critical  temperature,  the  liquid  being  identical  with  the 
vapour.  Since  this  temperature  is  practically  the  critical 
temperature  of  water  itself,  it  is  to  be  concluded  that  the 
solubility  of  the  salt  in  the  critical  vapour  is  practically  zero. 
The  point  P  in  fact  lies  quite  close  to  the  temperature  T  axis. 
(For  the  sake  of  clearness,  the  critical  point  has  been  drawn  at 
too  great  a  concentration,  because  otherwise  one  could  not 
show  that  in  P  the  (solubility  melting  point)  line  passes  con- 
tinuously into  the  vapour  line  PH,  which  has  been  assumed  as 
coinciding  with  the  axis  for  H2O°  below  320  C.) 

THE  TWO-COMPONENT  SYSTEM  FERRIC  CHLORIDE — WATER. 

This  system  is  of  interest,  as  it  was  the  first  case  of 
systematic  examination  of  the  hydrated  salts  from  the  stand- 
point of  the  Phase  Rule,  undertaken  by  Bakhuis  Roozeboom. 
The  diagram  (Fig.  83),  in  which  the  ordinates  represent 
temperature  and  the  abscissae  molar  concentration  of  ferric 
chloride  (reckoned  as  Fe2Cl6)  per  100  moles  of  water,  will 
illustrate  the  behaviour  of  this  system.  It  will  be  seen  that  the 
1  Nacken,  Smits  and  Wuite,  I.e. 


TWO   COMPONENT  SYSTEMS 


291 


following  compounds  are  marked  :  Fe2Cl6i2H2O,  Fe2Cl67H2O, 
Fe2Cl65H2O,  Fe2Cl64H2O,  Fe2Cl6  anhydrous.  The  discovery 
of  these  various  compounds  has  been  made  possible  by  the 
application  of  the  Phase  Rule.  In  fact,  the  usefulness  of  the 
Phase  Rule  appears  in  many  cases  in  which  it  is  quite  im- 


Fe2Cl6l2aq. 


FezCI6  5aq. 


I2H20  7HS0  5H20  4-H20 

Composition 


0  (O  20  30 

Moles.Fe2  CL6/WO  moles.H2  0. 
FIG.  83. 

possible  to  isolate  a  given  compound,  yet  the  behaviour  of  the 
system  indicates  without  doubt  that  such  a  compound  exists. 
Let  us  begin  by  considering  pure  water  (freezing  point  o°  C., 
point  A,  lower  half  of  diagram)  to  which  we  add  increasing 
quantities  of  anhydrous  ferric  chloride.  Measuring  successive 
freezing  points  we  pass  along  the  line  AB.  At  the  point  B,  the 


292        A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

system  will  go  completely  solid,  and  microscopic  analysis  shows 
that  there  is  a  heterogeneous  eutectic  mixture  formed  at  B,  the 
phases  being — solid  ice,  solid  Fe2Cl6i  2H2O — solution — vapour. 
By  raising  the  temperature  and  adding  more  ferric  chloride  we 
obtain  the  solubility  curve  BC,  which  passes  through  a  maxi- 
mum, and  again  another  eutectic  mixture  is  formed  at  D. 
From  the  shape  of  the  curve  BCD,  one  showing  a  maximum 
falling  to  a  eutectic  on  each  side,  we  infer  the  existence  of  a 
compound  represented  by  the  point  C.  This  point  corresponds 
to  the  formula  Fe2Cl6i2H2O.  The  solution  has  this  composi- 
tion, that  is,  the  solution  is  really  the  fused  solid.  In  other 
words,  if  we  choose  a  solution  having  the  composition  indicated 
by  C  and  slightly  lower  the  temperature,  it  will  be  found  that 
the  whole  system  will  solidify  sharply  at  C,  without  change  in 
temperature,  thereby  indicating  the  existence  of  a  true  com- 
pound, a  conclusion  which  is  supported  by  taking  the  shape 
of  the  curve  into  consideration.  By  the  successive  addi- 
tion of  salt  and  raising  or  lowering  the  temperature  we  can 
trace  out  the  remainder  of  the  diagram,  which  shows  further 
eutectics  at  the  points  F,  H,  K,  and  also  indicates  the 
compounds  at  the  points  E,  G,  I.  If  a  horizontal  line  be 
drawn  just  below  C,  as  indicated  in  the  figure,  it  will  be  seen 
that  the  solid  dodecahydrate  can  exist  in  equilibrium  at  one 
and  the  same  temperature,  with  two  solutions  of  quite  different 
concentrations  indicated  respectively  by  ;;/]_  and  M2.  The 
possibility  of  this  from  the  kinetic  standpoint  is  that  the  sort 
of  molecules  present  in  the  two  cases  is  different,  or  rather 
the  relative  proportions  of  certain  sorts  of  molecules  are 
different.  As  we  have  seen  in  dealing  with  the  case  of  solu- 
tions of  sodium  sulphate  we  must  consider  that  molecules  of 
all  the  compounds  are  present,  though  at  low  temperatures 
and  concentrations,  say  in  the  region  of  mlt  there  will  be 
exceedingly  few  of  the  anhydrous  Fe2Clg  molecules  or  of  the 
tetrahydrate  or  pentahydrate,  more  of  the  heptahydrate,  and 
a  great  many  dodecahydrate  molecules.  In  the  concentration 
region  ;;/2,  there  will  be  less  dodecahydrate  and  a  greater 
number  of  heptahydrate  molecules.  The  equilibrium  between 
all  these  different  sorts  of  molecules  in  the  homogeneous 


TWO   COMPONENT  SYSTEMS  293 

solution  is  presumably  governed  by  the  law  of  mass  action. 
Owing  to  the  continuous  change  in  the  molecular  nature  of  the 
solution  with  rising  concentration  and  temperature,  we  must 
not  regard  the  solution  m%  as  simply  a  more  concentrated  form 
of  mlt  for  if  we  did  we  would  get  the  thermodynamically  im- 
possible case  of  one  and  the  same  substance  (solid  dodeca- 
hydrate)  existing  in  equilibrium  with  the  same  solution  at 
different  concentrations. 

If  now  we  wish  to  isolate  or  prepare  a  given  hydrate  of 
ferric  chloride  all  we  have  to  do  is  to  examine  the  diagram 
and  note  the  concentration  of  solution  and  the  temperature  at 
which  a  pure  hydrate  solidifies,  i.e.  any  of  the  points  C,  E, 
G,  I.  As  a  practical  guide,  especially  in  technical  practice 
such  as  in  chemical  manufacture,  the  Phase  Rule  is  of  great 
assistance. 

The  system  ferric  chloride — water  may  be  used  to  illustrate 
an  important  point,  namely,  the  process  of  dehydration  of  a  solid 
hydrate.  Each  solid  hydrate  has  a  certain  pressure  of  water 
vapour  with  which  it  is  in  equilibrium.  The  upper  half  of 
the  figure  shows  diagrammatically  the  values  of  such  pressures, 
called  dissociation  pressures,  as  a  function  of  composition,  the 
temperature  being  constant.  Consider  what  happens  if  water 
be  removed,  by  isothermal  distillation  or  desiccation,  from  a 
hydrate  of  high  water  content  (say  the  dodecahydrate).  The 
diagram  shows  that  the  pressure  will  not  alter  continuously  as 
distillation  proceeds,  but  will  alter  discontinuously,  remaining 
constant  for  a  time,  at  a  series  of  different  values  corresponding 
to  the  dissociation  pressures  of  successive  hydrates. 

It  is  very  important  to  note  that  "  the  dissociation  pressure 
of  a  hydrate  "  is  really  the  equilibrium  pressure  over  two  salts 
simultaneously.  The  vapour  consists  of  water  molecules,  since 
those  of  the  salt  are  practically  non- volatile.  The  "  dissociation 
pressure  of  a  given  hydrate "  (say  the  dodecahydrate)  might  be 
regarded  as  identical  with  the  "  true  "  vapour  pressure  of  the  next 
lower  hydrate  (the  heptahydrate),  as  far  as  affinity  processes 
are  concerned,  cf.  Chap.  XI.  The  dodecahydrate  exists  in 
equilibrium  with  solid  heptahydrate  and  water  vapour  at  the 
dissociation  pressure  of  the  dodecahydrate.  (Naturally  the 


294       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

production  of  watef  molecules  from  the  salt  must  have  de- 
hydrated a  small  portion  of  it  at  least  to  the  next  lower 
hydrate.)  We  have  thus  a  two-component  system  existing  in 
three  phases,  and  as  we  have  assumed  the  temperature  constant, 
it  follows  that  the  pressure  must  be  fixed,  namely  the  dissocia- 
tion pressure.  On  continuing  the  desiccating  process,  solid 
heptahydrate  is  produced  at  the  expense  of  the  dodecahydrate 
at  constant  temperature  and  pressure,  until  finally  the  last  trace 
of  dodecahydrate  disappears,  the  pressure  being  still  the  same, 
and  the  solid  being  the  heptahydrate.  Such  a  state  of  things, 
however,  is  only  a  limiting  case,  for  if  the  smallest  quantity  of 
water  is  further  removed  some  pentahydrate  is  formed,  and 
the  pressure  falls  suddenly  to  the  "dissociation  pressure"  of 
heptahydrate.  The  same  change  is  observed  when  the  hepta- 
body  entirely  disappears,  and  the  solid  system  consists  of 
penta-  and  tetrahydrate,  a  further  change  being  observed  when 
the  pentahydrate  disappears  and  the  tetrahydrate  is  partly 
converted  into  anhydrous  salt.  The  water  vapour,  which  is  at 
the  dissociation  pressure  of  tetrahydrate,  is  in  equilibrium  with 
tetrahydrate  and  anhydrous  salt  simultaneously.  If  we  con- 
tinue removing  water  vapour  all  the  tetrahydrate  may  be  made 
to  vanish  at  constant  temperature  and  pressure,  and  we  find 
in  the  limit  that  the  anhydrous  salt  can  exist  in  equilibrium 
with  a  certain  pressure  of  water  vapour,  namely  "  the  dissocia- 
tion pressure  of  the  tetrahydrate."  The  distinction  between 
"  true "  vapour  pressures  over  a  solid  and  the  dissociation 
pressure  will  be  clear  when  we  come  to  study  affinity  of 
hydrate  formation. 

The  study  of  three  and  more  component  systems  would 
take  us  beyond  the  scope  of  this  book.  The;  reader  is  there- 
fore referred  to  the  special  works  and  papers  already  mentioned. 
This  chapter  will  be  concluded  with  a  brief  description  of 
Smit's  new  theory  of  allotropy,  which  is  of  fundamental 
importance. 


ALLOTROPY  295 


THE  PHENOMENON  OF  ALLOTROPY. 

Owing  to  the  amount  of  research  which  has  been  devoted 
to  this  subject  during  recent  years  it  is  necessary  to  briefly 
survey  some  of  its  more  important  features. 

When  one  and  the  same  compound  can  exist  in  two  or 
more  forms  (differing  in  crystalline  form  and  other  physical 
properties)  the  compound  is  said  to  exhibit  isomerism  or 
polymorphism.  In  the  case  of  some  elements,  such  as  sulphur, 
phosphorus,  selenium,  and  tellurium,  a  similar  phenomenon 
has  been  observed,  to  which  the  name  allotropy  has  been  given. 
There  are  now  three  types  of  allotropy  recognised.  First,  the 
allotropy  may  be  Enantiotropic,  the  different  varieties  of  an 
element  possessing  definite  temperature  ranges  of  stability 
and  convertible  one  into  the  other  at  a  certain  temperature 
and  pressure  called  the  transition  point.  An  example  is  the 
grey  and  white  tin  transformation  (Cohen,  Lc.)  which  has 
already  been  referred  to.  Secondly,  the  allotropy  may  be 
Monotropic,  that  is  one  variety  is  perfectly  unstable  at  all 
temperatures  and  pressures.  The  unstable  form  tends  to  pass 
continuously  into  the  stable.  An  example  of  this  is  explosive 
antimony,  investigated  by  Cohen  and  his  collaborators. 
Thirdly,  we  have  Dynamic  allotropy.  In  this  case  the  different 
varieties  or  allotropes  can  exist  together  in  certain  porpor- 
tions,  there  being  an  equilibrium  between  them  (presumably 
governed  by  the  law  of  mass  action),  such  equilibrium  points 
being  shifted  by  change  in  temperature.  The  phenomenon  of 
dynamic  allotropy  is  quite  analogous  to  that  of  dynamic 
isomerism,  except  that  in  the  first  the  substance  is  an  element, 
in  the  second  the  substance  is  a  compound. 

The  first  case  of  dynamic  allotropy,  namely,  that  of  liquid 
sulphur,  was  investigated  by  Alexander  Smith  and  his  collabo- 
rators (Journ.  Amer.  Chem.  Soc.,  1903,  and  onwards),  who 
showed  that  liquid  sulphur  is  a  mixture  of  a  light  mobile 
variety  soluble  in  carbon  bisulphide  and  denoted  by  the 
symbol  SA,  together  with  a  dark  viscous  variety  insoluble  in 
carbon  bisulphide  denoted  by  S^.  Ordinary  liquid  sulphur  at 


296       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

the  "  natural  freezing  point,"  H4'5°  C.,  consists  of  3-6  per  cent. 
S/i,  the  remainder  being  S*.  The  freezing  point  is,  however, 
characterised  by  the  phenomenon  of  variability  according  to 
the  composition  of  the  liquid,  i.e.  according  to  the  amount  of 
Sfj.  present,  and  according  to  the  solid  phase  separating  out. 
The  following  data  are  given  by  Smith  and  Carson  (Zeitsch. 
physik.  Chem.,  77,  661,  1911)  for  the  "ideal  freezing  points" 
when  no  S/i  is  present,  and  the  "  natural  freezing  points  "  when 
the  Sp,  is  present  in  the  liquid  in  the  equilibrium  proportion. 

Ideal  F.P.  Natural  F.P. 

Rhombic  Sulphur  (82) ^19-25°  114*5° 

(3'6%  SM) 

Prismatic  (Monoclinic)  Sulphur  (S2)  112-8°  no'20 

(3*4%  SM) 

Nacreous  Sulphur T  (S3)  ....  io6'8°  103-4° 

(3-i%  SM) 

The  phenomenon  of  allotropy  has  recently  been  investigated 
and  the  theoretical  views  extended  by  Smits  and  his  co-workers 
(A.  Smits,  Zeitsch.  physik.  Chem.,  76,  421,  1911;  ibid.,  77, 
367,  1911;  also  Proc.  Roy.  Soc.  Amsterdam^  12,  763,  1909- 
1910;  ibid.,  13,  822,  1911;  ibid.,  14,  1199,  1912).  Smits 
emphasises  the  idea  that  not  only  does  the  liquid^  i.e.  the 
fused  element,  contain  molecules  of  different  sorts,  but  that  the 
solid  separating  is  likewise  a  solid  solution,  the  inner  equili- 
brium existing  in  the  liquid  state  having  its  counterpart  in  the 
solid  state  as  well.  Following  out  this  idea,  Smits  has  shown 
the  relation  which  exists  between  the  three  kinds  of  allotropy, 
for  a  consideration  of  which  the  reader  is  referred  to  the 
original  papers.  Before  considering  the  case  of  phosphorus, 
which  we  shall  take  as  an  illustration  of  Smit's  method  of 
treatment,  it  is  necessary  to  consider  the  means  whereby  the 
existence  of  inner  equilibrium  between  different  sorts  of  mole- 
cules can  be  inferred. 

NOTE. — A  unary  substance  is  one  whose  molecules  are  all 

1  This  variety  may  be  prepared  in  needle-shaped  crystals  by  heating 
sulphur  to  150°  C.,  cooling  to  98°  C.,  and  making  it  crystallise  by 
scratching. 


ALLOTROPY  297 

identical  physically  as  well  as  chemically.  A  pseudo-binary 
substance  is  one  whose  molecules  are  chemically  the  same 
(as  regards  ultimate  analysis),  but  nevertheless  may  be  divided 
into  two  sorts  differing  from  one  another  in  respect  of 
"  physical  "  properties,  and  there  exists  an  equilibrium  between 
the  two  sorts. 

The  Significance  of  Melting-Point  Determinations  from  the 
Standpoint  of  the  Theory  of  Allotropy. 

Consider  the  diagram  (Fig.  84),  in  which  the  ordinate 
denotes  temperattire^  and  the  abscissae  time.  In  the  case  of 
an  absolutely  unary  body  con- 
sisting of  molecules  of  only  one 
sort,  both  in  the  liquid  and 
solid  states,  the  curve  obtained 
would  be  similar  to  that  shown 
with  a  perfectly  horizontal  por- 
tion  during  the  process  of 
solidification.  ABCD  repre- 
sents the  cooling  curve,  i.e.  the 


temperature-time   curve,   which  ^ 

we  get  when  a  substance  be- 

haves as  a  perfectly  unary  substance^  and  the  heterogeneous 
equilibrium  (i.e.  transformation  of  phase  at  constant  tem- 
perature) between  the  liquid  substance  surrounding  the 
immersed  thermometer  and  the  outer  solidifying  layer  (which 
is  in  contact  with  the  cooling  bath)  sets  in  rapidly  enough 
for  the  loss  of  heat  to  be  compensated  by  the  heat  evolved 
by  the  process  of  crystallisation.  The  rounding  at  B  is 
due  to  the  fact  that  the  thermal  conductivity  of  the  liquid 
is  not  perfect,  and  therefore  thermal  equilibrium  between 
outer  and  inner  parts  cannot  be  established  instantaneously. 
Solidification  in  the  outer  layers  begins  before  the  liquid  in 
the  immediate  neighbourhood  of  the  thermometer  bulb  has 
fallen  to  the  temperature  of  solidification.  The  flat  part  BC  is 
due  to  heat  compensation.  That  is  when  the  liquid  round  the 
thermometer  has  reached  the  temperature  of  solidification, 


298       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

the  heat  lost  to  th^bath  is  balanced  by  heat  evolved  on 
crystallisation,  and  the  horizontal  part  of  the  curve  results. 
Before  the  mass  has,  however,  become  entirely  solid,  a  change 
sets  in,  because  the  thermometer  comes  more  and  more  in 
contact  with  the  solid  substance,  which  in  so  far  as  the  bulb 
is  not  in  direct  contact  with  the  liquid  will  possess  a  lower 
temperature  than  the  liquid,  and  will  produce  in  its  readings 
the  rounding  at  C,  until  the  last  trace  of  liquid  has  vanished. 
If  the  heterogeneous  equilibrium,  i.e.  the  transformation  of  solid 
into  liquid  at  constant  temperature,  is  not  established  with 
very  great  rapidity,  the  loss  of  heat  can  no  longer  be  compen- 
sated by  the  heat  of  crystallisation,  in  consequence  of  which 
the  liquid  in  contact  with  the  solid  is  super-cooled.  In  this 
case  a  more  or  less  descending  line  will  be  found  instead  of  the 
horizontal  part.  Besides  this  lag  or  hysteresis  in  the  hetero- 
geneous equilibrium  solid  ^liquid  causing  a  sloping  freezing 
curve,  it  is  also  conceivable  that  the  supercooling  might  be 
attributed  to  the  fact  that  the  homogeneous  inner  equilibrium 
between  different  sorts  of  molecules  of  the  same  compound, 
has  not  set  in  quickly  enough,  i.e.  the  substance  is  not  acting 
as  a  unary  one.  Accordingly,  to  decide  by  means  of  cooling 
curves  whether  or  no  a  substance  behaves  in  a  unary  way,  we 
must  pursue  the  following  course :  during  the  actual  cooling 
curve  the  circumstavices  are  made  as  similar  as  possible  (in  a 
series  of  experiments),  whereas  the  previous  history  of  the 
(liquid)  substance  is  made  as  different  as  possible,  i.c.  it  is  raised 
to  different  temperature  stages,  and  quickly  brought  down  to 
the  freezing  point,  the  idea  being  that  the  inner  equilibrium 
will  thus  vary  from  case  to  case,  and  will  manifest  itself  by 
different  forms  of  solidifying  curves,  the  heterogeneous  equili- 
brium changes  being  presumably  kept  the  same  in  successive 
experiments  by  freezing  at  the  same  rate,  etc.  In  the  actual 
case  the  liquid  substance  is  first  allowed  to  supercool  a  little, 
and  is  then  "  seeded  "  in  some  way,  because  the  maximum  to 
which  the  temperature  then  rises  in  the  subsequent  solidifica- 
tion can  give  valuable  information  regarding  the  existence  or 
non-existence  of  inner  equilibrium,  i.e.  as  regards  settling 
whether  a  substance  is  unary  or  not.  Smits  has  examined  in 


ALLOTROPY  299 

this  way  the  systems :  Mercury  (which  behaved  as  a  unary 
substance,  and  may  therefore  be  regarded  as  composed  of 
identical  molecules);  tin  (which  proved  to  be  complex, 
possibly  pseudo-binary,  i.e.  two  sorts  of  molecules)  ;  water 
(which  also  proved  to  be  complex,  the  "  freezing  point " 
varying  from  — 0*28  to  — o'o6  when  solid  ice  was  rapidly 
heated);  and  finally  sulphitr  and  phosphorus •,  the  latter  of 
which  will  be  now  considered  briefly. 

Phosphorus  probably  exists  in  three  solid  forms  as  well  as 
liquid  and  gaseous  (Jolibois,  Comptes  Rendus,  149,  287,  1909; 
151,  382,  1910) — white  phosphorus,  red,  and  pyromorphic  or 
violet  phosphorus.  After  considerable  trouble  perfectly  pyre 
white  phosphorus  was  obtained,  i.e.  white  phosphorus  which 
on  being  heated  slowly  gave  a  sharp  melting  point.  It  was 
found  to  be  44*0°  C.  Having  thus  an  apparent  unary  behaviour 
as  far  as  slow  temperature  changes  are  concerned,  the  next 
thing  was  to  see  if,  by  rapid  heating  or  cooling,  the  real  inner 
complexity  would  manifest  itself.  The  following  is  a  descrip- 
tion of  three  experiments.  The  melting  point  vessel  was  first 
placed  in  boiling  water  for  some  time  and  then  suddenly  trans- 
ferred to  a  bath  at  15°  C.  to  make  the  cooling  take  place  so 
rapidly  that  the  internal  equilibrium  could  not  keep  pace  with 
it.  When  the  grafting1  took  place  at  about  43*5°  C.,  after 
taking  out  of  the  bath,  the  temperature  rose  above  44°  C., 
from  which  it  followed,  therefore,  that  when  the  cooling  takes 
place  very  rapidly  the  liquid  phosphorus  is  already  supercooled 
at  44°  (its  true  unary  melting  point).  In  a  second  case  the 
grafting  took  place  above  44°  C.  and  the  temperature  rose  to 
45-5°,  and  Smits  succeeded  in  getting  a  rise  to  46°  with  grafting 
at  a  still  earlier  stage.  In  the  experiment  corresponding  to 
the  curve  given  (Fig.  85)  grafting  took  place  at  about  44*5°  C.; 
at  first  the  temperature  descended,  then  rose  to  45*05°,  after 
which  it  fell  again,  at  first  pretty  rapidly,  then  less  rapidly,  and 
at  last  very  rapidly  again.  The  whole  line  shows  the  type  of 
a  line  of  solidification  of  a  mixture,  the  melting  range  being 

1  The  grafting  was  effected  by  breaking  off  the  capillary  ending  to  the 
tube  and  inserting  it  for  a  moment  in  solid  CO2.  Solid  phosphorus  was 
thereby  formed. 


300       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 


here  about  i'8°,  but  it*can  be  considerably  larger  still.  When 
solid  phosphorus,  even  when  rapidly  cooled  so  as  to  exhibit 
the  phenomena  just  described,  is  allowed  to  stand  for  a  short 
time  inner  equilibrium  rapidly  sets  in.  The  second  curve 
(Fig.  86)  is  a  heating  curve  which  shows  what  was  observed 
after  the  solid  substance  obtained  in  the  previous  experiment 

was  suddenly  placed  in 
a  bath  at  50°  C.  The 
heating  curve  shows  that 
after  some  minutes  a 
considerable  approach 


45 

T 

44° 

3*3' 
«} 

5 


40° 


,4-5-05 


'•^4-3-25' 


4397° 


44 -to' 


Phosphorus 
FIG.  85. 


t-    time 


Time 


Phosphorus 
FIG.  86. 


to  the  state  of  internal  equilibrium  has  taken  place  but  has  not 
been  reached  as  yet,  for  the  melting  range  still  amounts  to 
0*13°  C.  and  the  end  melting  point  lies  above  the  unary 
melting  point.  Now  Smits  assumes  that  phosphorus  really 
possesses  two  kinds  of  molecules  mutually  convertible.  He 
denotes  these  hypothetical  "  forms  "  of  phosphorus  by  Pa  and 
Pp.  The  actual  forms  we  meet  with  (white,  red,  and  violet) 
are  really  solid  solutions  of  Pa  and  P^,  and  differ  from 
one  another  in  their  percentage  composition  in  respect  of 
these  constituents.  Of  course  it  could  scarcely  be  hoped  to 
ever  realise  experimentally  pure  100  per  cent.  Pa  or  TOO  per 


ALLOTROPY 


301 


cent.  P/j,  because  of  the  rapid  change  each  of  these  would 
undergo  into  some  known  form  where  a  and  j3  molecules  are 
both  present.   Smits  takes 
Pa  to  be  "  probably  colour- 
less."     This  seems  to  be 
necessary  because   Chap- 
man (Trans.  Chem.  Soc., 
75,  743,  1899)  found  that 
red  phosphorus  melts  to 
a   colourless    liquid    and 
in  becoming  liquid  there 
must  have   been   a   shift 
towards  one  or  other  of 
the   hypothetical  forms — 
we  may  take  this  one  to 
be  Pa.     The  behaviour  of 
the  system  phosphorus,  so 
far  as  it  is  known  to  us, 
could  according  to  Smits 
be   accounted   for   if  we 
take   the   various   phases 
known   to   be   related  to 
one  another  in  terms    of 
a  and  j8  (as  regards  their 
constitution  and  range  of 
stability),  as  given  in  the 
accompanying       diagram 
(Fig.  87).    The  line  K/2/i 
denotes  the  internal  equi- 
librium    in     the     liquid 
at  different  temperatures, 
and    s\)i    refers    to    the         Pa  *    P/3 

internal  equilibrium  (over  Phosphorus 

a       small       temperature 

range)  in  solid  white  phosphorus,  so  that  s±  and  /j_  indicate 
the  solid  and  the  liquid  phases  which  are  in  internal  equi- 
librium and  coexist  at  the  unary  melting  point  of  white 
phosphorus  (44°  C.).  Now  it  follows  from  the  course  of  the 


302       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

lines  that  if  the  liquid  /0  is  cooled  very  rapidly,  the  system 
will  move  down  the  dotted  vertical  line  and  crystallisation  will 
occur  already  at  l\  (the  composition  of  the  solid  being  s\). 
Then,  in  the  absence  of  internal  transformation,  a  melting  range 
l\l\  would  be  found,  whereas  in  the  case  of  rapid  heating  of  the 
solid  phase  ;/  the  melting  will  already  begin  at  s'2  and  be  com- 
pleted at  sv  Internal  conversions,  however,  are  not  absent, 
so  that  "  a  transgression  of  the  unary  melting  point  is  always 
much  smaller  than  the  lines  /Vi  and  j'2Ji  would  lead  us  to 
expect."  Further,  the  figure  shows  that  the  initial  solidification 
will  appear  the  sooner  according  as  a  higher  temperature  is 
started  from,  the  system  passing  down  (if  cooled  quickly)  an 
imaginary  vertical  line  from  any  point  on  the  /£/2/i  line.  Simi- 
larly we  see  that  initial  melting  will  appear  the  sooner  (i.e.  at  a 
temperature  lower  than  44°)  the  lower  the  temperature  we  start 
from.  All  this  is  to  be  expected  on  the  theory  of  allotropy, 
treating  phosphorus  as  a  pseudo-binary  system.  Jolibois  has 
remarked  that  violet  phosphorus  is  stable  below  460°  C.  and 
above  this  temperature  the  red  modification  is  stable,  which 
melts  to  a  colourless  liquid  at  6io°C.  Smits  points  out  that  a 
pseudo-binary  system,  i.e.  a  system  consisting  of  two  sorts  of 
molecules,  can  be  made  to  account  for  the  existence  of  three 
actual  crystalline  modifications  provided  that  it  is  assumed  that 
there  is  a  second  discontinuity  in  the  series  of  solid  solutions. 
This  has  been  done  in  the  figure.  The  liquids  along  bl  coexist 
with  the  solid  solutions  ds't  the  liquids  along  Ic  coexisting  in 
equilibrium  with  the  solid  solutions  se.  In  this  figure  s2 
represents  the  unary  melting  point  of  red  phosphorus,  the 
liquid  in  equilibrium  being  represented  by  /2  (temperature  610°). 
Below  this  temperature  red  phosphorus  remains  stable  to  460° 
(i.e.  the  system  cooled  slowly  will  pass  from  s2  to  ss).  At  460° 
the  red  s3  is  converted  into  violet  s±,  which  is  therefore  stable 
below  460°  C. 

It  must  be  remembered  that  a  considerable  part  of  the 
above  equilibria  at  the  present  time  are  of  a  hypothetical 
nature  only.  Other  views  on  the  composition  of  the  phosphorus 
system  are  held,  for  example,  by  Cohen  and  Olie  (Zeitsch. 
physik.  C/ie/H.,  71,  i,  1910). 


CHAPTER   X 

Chemical  equilibrium  in  heterogeneous  systems  (from  the  thermodynamic 
standpoint)  when  capillary  or  electrical  effects  are  of  importance — 
Adsorption — Donnan's  theory  of  membrane  equilibria. 

ADSORPTION 

THE  phenomena  with  which  we  have  to  deal  are  those  which 
are  manifested  at  the  interface  where  two  phases  meet.  The 
general  treatment  of  the  behaviour  of  heterogeneous  systems 
in  which  surface  or  interface  effects  due  to  capillarity  have  to 
be  taken  into  account  (as  for  example  the  stability  of  colloidal 
solutions  and  emulsions),  from  the  standpoint  of  a  modified 
Phase  Rule,  has  been  left  so  far  almost  untouched.  Within  recent 
years  Pawlow  (Zeitsch.  physik.  Chem.^  75,  48, 1910)  seems  to  be 
the  only  investigator  to  have  attempted  this  difficult  problem, 
but,  rather  remarkably,  his  work  has  received  little  or  no  atten- 
tion. Willard  Gibbs  himself  pointed  out  in  his  original  memoirs 
on  "  Equilibrium  in  Heterogeneous  Systems  "  (Scientific  Papers^ 
vol.  II.)  that  in  addition  to  the  variables  already  considered 
in  the  Phase  Rule,  namely,  temperature,,  pressure,  and  con- 
centration, one  must  also  take  into  account  the  surface  area 
of  the  interface  (or  interfaces).  The  difficulty  is  to  settle  the 
correct  number  of  equations  connecting  these  variables. 

Donnan  has  investigated  the  problem  of  stability  of  colloidal 
solutions  from  the  thermodynamic  standpoint,  starting  with 
the  idea  of  an  effective  negative  surface  tension.  His  in- 
vestigation, which  is  an  extension  of  that  given  in  Part  I. 
(Vol.  I.)  of  this  book,  will  be  found  in  the  Zeitsch.  physik. 
Chem.,  46,  197,  1903. 

Instead  of  pursuing  so  general  a  method  of  treatment,  the 
subject  of  capillary  chemical  effects  has  been  studied  ex- 
perimentally, by  investigations  of  Gibbs'  expression  for  the 
surface  concentration  effects  due  to  surface  tension.  Gibbs 


304       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

(I.e.)  showed  as  a  t^ermodynamical  necessity  that  IF  A  DIS- 
SOLVED SUBSTANCE  HAD  THE  PROPERTY  OF  LOWERING  THE 

SURFACE  TENSION  OF  THE  SOLUTION  (SAY,  AT  THE  LIQUID  |  AIR 
SURFACE),  THE  SUBSTANCE  WOULD  EXIST  AT  A  HIGHER  CON- 
CENTRATION IN  THE  SURFACE  LAYER  THAN  IN  THE  BULK  OF 
THE  SOLUTION. 

This  surface  concentration  is  identified  with  the  phenome- 
non of  Adsorption,  although  it  must  be  remembered  that  effects 
which  are  usually  described  under  this  title  partake  more 
frequently,  in  part  at  least,  of  partial  solution  of  the  substance 
in  the  second  phase,  and  also  possibly  include  some  kind  of 
purely  chemical  change,  not  taken  account  of  by  the  simple 
physical  theory.  The  property  possessed  by  charcoal  of 
removing  substances  (such  as  colouring  matter)  from  solution, 
or  "  absorbing  "  gases  and  vapours,  is  certainly  an  example  of 
adsorption  purely  physical  in  the  first  place  (and  perhaps 
entirely),  but  possibly  also  partaking  of  the  nature  of  solubility, 
such  as  that  to  which  the  Distribution  Law  applies.  The 
process  of  dyeing  is  also  an  instance  of  physical  adsorption, 
followed  in  many  cases  by  chemical  changes  in  the  dye  itself 
or  even  in  a  chemical  reaction  between  the  dye  and  the 
material  dyed.  The  fastness  of  dyes,  the  greatest  desideratum 
from  the  technical  standpoint,  means  simply  the  irreversibility 
of  the  process  and  this  is  in  itself  sufficient  to  show  that 
dyeing  cannot  be  entirely  due  to  physical  adsorption  dealt 
with  in  the  Gibbs'  theory,  since  it  is  explicitly  assumed  in  the 
latter  that  the  surface  concentration  effects  are  reversible. 
Investigation  of  the  Gibbs'  expression  must  therefore  be  made 
under  conditions  where  chemical  effects  and  solubility  of  the 
solute  in  the  adsorbing  phase  (the  charcoal,  for  example)  are 
reduced  to  a  minimum.  There  can  be  no  doubt  that  more 
complete  information  of  this  purely  physical  phenomenon  is 
the  first  step  towards  a  rational  understanding  of  the  more 
complicated  phenomena  of  the  dye-house  and  the  filter  bed. 

Since  the  Gibbs'  adsorption  equation  (as  we  shall  call  it) 
is  thus  of  very  considerable  importance  for  capillary  chemistry, 
the  deduction  of  the  equation  by  means  of  a  thermodynamical 
cycle  will  not  be  without  interest. 


GIBBS  EQUATION  305 

Deduction  of  the  Adsorption  Equation. 

The  following  is  the  deduction  given  by  Freundlich 
(Kapillarchemie^  p.  50). 

Consider  a  dilute  solution  of  volume  z/,  osmotic  pressure  P, 
in  contact  with  a  vapour  phase  consisting  only  of  the  vapour 
of  the  solvent.  The  area  of  the  separating  surface  between 
liquid  and  vapour  is  s,  and  the  surface  tension  is  <r.  The 
solution  is  placed  in  a  vessel  fitted  with  a  piston  having  a 
semi-permeable  membrane,  the  solution  being  on  one  side  of 
the  membrane,  and  on  the  other  side  an  infinite  reservoir  of 
pure  solvent  in  contact  with  it.  The  following  cyclic  process 
is  carried  out. 

The  surface  area  is  increased  by  the  amount  dst  the  work 
done  being  —  vds.  The  volume  of  the  solution  is  considered 
as  having  remained  constant.  At  the  same  time  the  osmotic 
pressure  P  may  have  altered — its  new  value  being  given  by  the 

expression  ( P  +  ~ ds\  The  volume  of  the  solution  is  now 
increased  by  dv,  by  pulling  out  the  piston,  the  work  being 

4-(p.f  -7T-  ds  \dv.     The  surface  area  s  is  supposed  to  have 
V          ds     ) 

remained  constant  in  this  last  operation  whilst  the  tension  a- 
has  changed  to  the  value  ( a-  -\-  v-  dv\  The  surface  area  now 
contracts  to  its  initial  value,  the  work  gained  being 
-j-  (fj-  _|_  ^  dv  \ds.  P  has  now  returned  to  its  initial  value,  and 

when  the  piston  is  pushed  in,  thereby  doing  the  work  —  P^z/, 
the  system  has  returned  to  its  initial  state.  Since  the  process 
is  isothermal  and  reversible  the  total  work  is  zero.  That  is — 


—<rds  +    P  +  IT**  +    o-  +  ^  —  P^P  =  o 

da  dP 


This  equation  states  that  if  the  surface  tension  alters  with  the 
volume,   that   is   with   the   concentration,    then   the    osmotic 
T.P.C.  —  ii.  x 


306       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

pressure  must  alter  with  the  surface  area.  The  latter  can  only 
be  the  case  if  the  concentration  of  the  solute  in  the  bulk  of 
the  solution  depends  on  the  surface  area,  and  this  can  only 
be  so  if  the  concentration  in  the  surface  layer  is  greater  or 
less  than  that  in  the  bulk  of  the  solution. 

From  the  above  we  see  that  the  concentration  c  of  the 
solution  is  a  function  of  v  (volume)  and  also  of  s  (surface 
area).  If  n  moles  are  dissolved,  then  we  cannot  simply  write 

<:  =  -,  but   instead  we  must   allow  for  the  fact  that  in  the 

surface  layer  the  solute  is  present  to  a  greater  or  less  extent 
than  in  the  bulk.  Suppose  we  denote  this  positive  or  negative 
excess  in  the  surface  layer  by  F,  where  T  is  mass  of  solute 
reckoned  per  unit  area  of  surface.  Then  if  the  surface  area  is 
5,  the  quantity  of  solute  in  excess  in  the  surface  layer  is  IV 
and  hence  the  actual  concentration  in  the  bulk  of  the  solution 

n  —  Ts 
is  given  by  c  =  --  .     Note  that  the  quantity  Ts  may  be 

positive  or  negative.  We  can  now  rewrite  the  above  equation 
(i)  in  the  form— 

do-    dc  _       dP   dc 

dc'fo"    ~dc'ds    .....     (2) 

But  dc-  H*-*) 

dv~      -v*~ 

dc        r 

and  d^~v 

so  that  equation  (2)  becomes  — 

da-  dP 


Since  the  solution  is  a  dilute  one  we  can  apply  the  gas  law 
P  =  RT<r,  so  that  we  finally  obtain  — 

r_       _£_  <?°" 
"RT'ar 

This  equation  states  that  if  the  surface  tension  decreases  as 
the  concentration  of  the  solute  increases,  then  F  is  positive,  that 
is  the  concentration  of  the  solute  in  the  surface  layer  is  greater 


GIBBS   EQUATION  307 

than  its  concentration  in  the  bulk.  This  is  positive  adsorption. 
On  the  other  hand,  if  the  surface  tension  increases  as  the 
concentration  increases,  there  will  be  a  negative  adsorption 
or  desorption  of  the  solute  in  the  surface  layer.  Finally,  if 
the  surface  tension  be  independent  of  concentration,  the  con- 
centration will  be  the  same  in  both  bulk  and  surface  layer. 
The  experimental  investigation  of  this  equation  was  first 
attempted  by  Donnan  and  W.  C.  McC.  Lewis  (cf.  Lewis, 
Phil.  Mag.,  1908  ;  ibid.,  1909).  Experiments  were  made  with 
aqueous  solutions  of  sodium  glycocholate  and  also  dyestuffs 
and  other  substances,  notably  caffeine,  which  exerted  by  their 
presence  considerable  lowering  upon  the  surface  tension  of 
the  water.  These  substances  likewise  exerted  a  considerable 
lowering  on  the  interfacial  tension  between  a  pure  hydrocarbon 
oil  and  the  aqueous  solutions.  It  was  shown  that  the  sub- 
stances did  not  dissolve  in  the  oil  and  no  chemical  action  was 
to  be  anticipated,  so  that  the  conditions  seemed  favourable 
for  the  verification  of  the  expression.  The  interfacial  tension 
between  the  oil  and  aqueous  solutions  of  various  concentra- 
tions was  measured,  the  tangent  to  the  curve  thus  obtained  at 

any  given  concentration  (c)  representing  the  value  of  -T-.     The 

right-hand  side  expression  could  thus  be  calculated.  In  the 
case  of  sodium  glycocholate  and  the  dyestuffs  it  came  out  to 
be  of  the  order  io~7  gram/cm.2.  The  value  of  the  left-hand 
side  was  directly  determined  by  two  different  methods.  In 
the  first  method  the  oil  was  emulsified  (i.e.  broken  up  into 
fine  droplets)  by  shaking  with  the  aqueous  solution,  the  change 
in  concentration  of  the  bulk  of  the  solution  being  determined 
(the  concentration-interfacial  tension  curve  was  itself  used  as 
the  analytical  means  as  it  was  the  most  delicate).  Only  a 
very  small  change  in  concentration  was  observed.  By  micro- 
scopic measurements,  the  size  of  the  oil  particles  was  deter- 
mined and  hence  their  number  and  hence  the  total  adsorbing 
surface  area.  Knowing  the  total  quantity  of  solute  removed 
from  the  bulk  of  the  solution,  and  the  total  absorbing  surface, 
one  obtains  directly  the  mass  adsorbed  per  cm.2.  The  order  of 
magnitude  in  the  case  of  these  substances  was  io-6  gram /cm.2. 


3c8       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

There  is  thus  a  lar£e  discrepancy  between  the  "  observed  " 
value  of  T  and  the  "calculated."     Thinking  that  the  great 
curvature  of  the  emulsion  particles  might  have  something  to 
do  with   this,  a  second  method  was   employed,  in  which  a 
stream  of  large  oil  drops  was  passed  for  a  long  time  through 
the  solution,  the  change  in  concentration  being  again  deter- 
mined.    The   results   came   out  almost   identical  with  those 
obtained  by  the  first  method.      With  the  caffeine,  however, 
although  the  quantity  adsorbed  was  much  less  than  in  the 
foregoing  cases,  approximate  agreement  was  obtained,  viz.  the 
observed  and  calculated  value  of  T  was  of  the  order   io~8 
gram /cm.2.     It   seems  likely,  therefore,  that  in  the  case  of 
the  dyestuffs  and  sodium  glycocholate  (which  exhibit  certain 
colloidal  properties)  the  discrepancy  is  due  to  colloidal  gela- 
tinisation  or  flocculation  on  the  oil  surface,  an  effect  which 
is   not    taken   account    of    in   the   simple   physical  relation- 
ship.    Later  measurements  by  Lewis  (Zeitsch.  physik.  Chem., 
73,  129,  1910)  on  the  adsorption  of  caffeine  by  mercury  (the 
caffeine  being  dissolved  in  aqueous  alcohol)  showed  agree- 
ment as  regards  order  of  magnitude.     Besides  the  measure- 
ments at   the  interface  of  two  liquids,  Donnan  and  Barker 
(Proc.  Roy.  Soc.,  85  A.,  557,  1911)  have  measured  the  adsorp- 
tion of  nonylic  acid  and  saponine  at  the  liquid  |  air  surface. 
The  air  was  passed  in  the  form  of  bubbles  through  a  column 
of  the  solution,  diaphragms  being  inserted  in  the  column  so 
as  to  avoid  general  stirring  of  the  liquid.     As  the  bubbles 
passed  up  they  carried  with  them  a  layer  of  higher  concen- 
tration of  the  nonylic  acid  than  that  of  the  bulk,  so  that  a 
decrease  in  concentration  of  the  nonylic  acid  in  the  lower 
parts   of  the   solution  occurred.     This   was  determined  and 
likewise  the  total  surface  area  of  the  adsorbing  bubbles,  and 
hence  the  value  of  T.     In  the  case  of  very  dilute  solutions  the 
adsorption  was  found  to  be  of  the  order  i  X  io~7  gram/cm.2 
for  nonylic  acid,  and  of  the  order  4  X  io~7  gram/cm.2  for 
saponine.     The  calculated  values  of  T  for  nonylic  acid,  namely 

the  expression  —  ^-^  ^r,  were  of  the  order  (0-26  —  o'63)io-7 
gram/cm.2,  the  quantity  adsorbed   increasing  with   the  bulk 


GIBBS  EQUATION  309 

concentration  of  the  solution.  In  the  case  of  saponine  the 
calculated  value  of  F  was  (1*36  —  r6o)  X  io~7  gram/cm.2. 
There  is  thus  satisfactory  agreement  between  observed  and 
calculated  values  at  least  as  far  as  order  of  magnitude  is 
concerned. 

A  considerable  difficulty  enters  if  we  have  to  try  and  allow 
for  electrocapillary  adsorption  as  well  as  pure  adsorption,  such 
as  probably  takes  place  in  the  case  of  mercury  salts  in  aqueous 
solution  in  contact  with  a  mercury  surface,  for  there  exists  here 
a  contact  difference  of  potential  which  modifies  the  value  of 
the  interfacial  tension  even  apart  from  concentration  of  the 
solute.  We  have  here  to  take  account  of  adsorption  of  ions 
as  well  as  adsorption  of  molecules.  Practically  nothing  is 
known  about  such  phenomena,  though  it  is  very  evident  that 
they  play  a  role — and  perhaps  a  fundamental  role — in  the 
mechanism  of  the  processes  involved  in  the  precipitation  of 
colloids  by  electrolytes. 

Donnan  has,  however,  been  very  successful  in  dealing  with 
a  problem  of  a  somewhat  similar  nature,  though  here  we  pass 
from  purely  one-sided  surface  effects  to  concentration  effects 
on  both  sides  of  an  interface.  The  problem  dealt  with  is  the 
distribution  of  ions  on  each  side  of  a  membrane,  the  ions 
being  due  to  an  "electrolytic  colloid,"  like  Congo  red  in 
water.  Congo  red  is  the  sodium  salt  of  an  organic  acid,  the 
molecules  and  anions  of  which  cannot  pass  through  the  mem- 
brane. This  is  of  importance  for  the  theory  of  dialysis  and 
colloids,  as  well  as  for  the  mechanism  of  living  plant  and 
animal  cells. 


DONNAN'S  THEORY  OF  "  MEMBRANE  EQUILIBRIA."  * 

Consider  a  salt  NaR  dissolved  in  water,  the  solution  being 
in  contact  with  a  membrane  (denoted  by  a  vertical  line)  which 
is  impermeable  to  the  anion  R'  and  also  to  the  undissolved 
molecules  NaR,  but  will  allow  Na"  and  any  other  ions  to  pass 
through  it  freely.  We  suppose  that  on  the  other  side  of  the 

1  Zeitsch.f.  Electrochemie,  17,  572,  1911. 


3io       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 


membrane  there  is  aik  aqueous  solution  of  NaCl.     The  initial 
state  of  things  will  be  represented  by 

Na'         Na' 

R'        cr 
i.        ii. 

The  NaCl  will  begin  to  diffuse  from  II.  into  I.  until  an  equi- 
librium state  is  reached,  represented  by 


Na' 
R' 

cr 
i. 


Na' 


cr 


Now  at  the  equilibrium,  if  a  small  virtual  change  is  made 
reversibly  at  constant  temperature  and  volume,  the  free  energy 
will  remain  unchanged,  i.e.  no  work  will  be  done.  The  change 
here  considered  is  the  transfer  of  8n  moles  of  Na'  and  Cl'  from 
II.  to  I.  The  work  which  we  set  equal  to  zero  is — 


=  o 


where  the  square  brackets  denote  concentration  terms.  It  is 
unnecessary  to  take  into  consideration  any  potential  difference 
which  may  exist  between  the  tadtsides  of  the  membrane,  since 
equivalent  quantities  of  posit^Kind  negative  electricity  have 
been  transferred  from  ^^^f>  If  we  carry  ont  an  exactly 
similar  work  proces&^flH  -  undissociated  NaCl  molecules 
from  II.  to  I. 


Tlog 


[NaCl]n 


" 


=  [NaCl]n 

ibining  this  equation  with  the  former  similar  relation  for 
the  ions,  we  obtain  — 

[NalfCI1] 

[NaCl] 
i.e.  the  Law  of  Mass  Action,  which  is  known  to  be  contrary 


MEMBRANE  EQUILIBRIA  311 

to  experience  (at  least  if  the  ordinary  conductivity  method  of 
determining  degree  of  dissociation  be  taken  as  giving  correct 
values).  Donnan  considers  the  discrepancy  may  be  due  to 
the  abnormality  of  the  undissociated  molecules  and  that  equi- 
librium across  the  membrane  need  not  necessarily  conform  to 
the  criterion  that  [NaCl]n  =  [NaCl]r  Returning  to  equation 
(i),  since  in  general  [Na']r  is  not  equal  to  [Na*]n,  because 
the  Na'  is  obtained  by  the  dissociation  of  both  NaR  and  NaCl, 
it  follows  that  [Cl']r  is  not  equal  to  [Cl']n.  To  get  at  some 
more  quantitative  relation,  one  may  make  the  following 
simplifying  assumptions  : — 

(a)  Complete  electrolytic  dissociation  of  NaR  and  NaCl. 

(b)  Equal  volumes  of  liquid  on  each  side  of  the  membrane. 
We  can  thus  represent  the  initial  and  equilibrium  states  as 

follows  : — 

Initial  state.  Equilibrium  state. 

Na'     R'         Na'     Cl'        Na*     R'   Cl'        Na'        Cl' 

i.  ii.        ||  i.  ii. 

where  the  symbols  c-^  cz  represent  gram-ions  per  liter.     That  is 
-loo  represents  the  percentage  of  NaCl  which  has  diffused 

from  II.  into  I.,  and  — the  equilibrium  distribution  ratio 

oc 

of  the  sodium  chloride  between  I.  and  II. 

Equation  (i)  can  now  be  written  in  the  form — 


or 
whence 


'2      fi  + 


x  cz 

If  c*  is  small  compared  to  clt  one  may  write- 

£  =  *     and     4±^  =  5 

c  2         Ci  X  C2 


312        A    SYSTEM  OF  PHYSICAL   CHEMISTRY 


By  way  of  illustration,  suppose  <r2  — — ~>  then  —  =  yoo>  or 


IOO 


only  i  per  cent,  of  the  NaCl  originally  present  in  II.  diffused 
into  I.  If  on  the  other  hand,  ^  is  small  compared  to  c^  it 
follows  that — 


>*,       i          , 
—  =  -      and 

'2         2 


=  i,  as  one  would  expect. 


The  following  table  shows  the  variation  of  the  distribution 
of  sodium  chloride  between  the  solutions  as  a  function  of  the 
concentration  of  the  NaR  and  NaCl  itself. 


Initial  concen- 
tration of  NaR 
in  I. 

Initial  concen- 
tration of  NaCl 
in  II. 

Initial  ratio 
NaR  |  NaCl. 

Percentage  of 
NaCl  diffused 
from  II.  to  I. 

> 

Distribution  ratio  of 
NaCl  between  II. 
and  I.  when  equi- 
librium is  reached. 

" 

* 

£ 

100    * 

.r    ' 

O'OI 
O'l 

I 

I 

I 
I 

O'l 

O'OI 
O'l 

I 

10 

497 
47'6 
33 

I'l 
2'0 
I  I'O 

I 

O'OI 

IOO 

I'O 

99-0 

The  equilibrium  reached  is  naturally  independent  of  the 
assumption  that  the  NaCl  was  initially  in  II.  The  same 
equilibrium  point  would  be  reached  if  the  NaCl  had  been 
initially  in  I.  The  table  shows  that  the  influence  of  the  non- 
dialysing  NaR  upon  the  distribution  of  the  NaCl  is  extremely 
great.  Although  the  membrane  is  quite  permeable  to  NaCl, 
the  presence  of  NaR  in  sufficient  concentration  on  one 
side,  is  able  to  make  the  permeability  of  the  membrane 
for  NaCl  in  one  direction  almost  vanish.  Donnan  points  out 
that  such  effects  must  be  of  great  importance  in  physiology, 
for  in  living  tissues  membranes  are  always  present  and  the 
existence  of  protein  salts  which  correspond  to  the  hypothetical 
substance  NaR  will  evidently  have  a  very  marked  effect  upon 
the  distribution  of  simple  inorganic  salts  to  which  these 


MEMBRANE  EQUILIBRIA  313 

membranes  are  '  '  normally  "  perfectly  permeable.  The  un- 
equal distribution  of  NaCl  on  two  sides  of  parchment  owing 
to  the  presence  on  one  side  of  congo  red,  has  been  experi- 
mentally demonstrated  by  Harris  in  Donnan's  Laboratory. 

Influence   of  the  Unequal  Distribution  on  the  Measurement  of 
Osmotic  Pressure. 

It  follows  from  what  has  been  said  that  no  direct  measure- 
ment of  the  osmotic  pressure  of  NaR  can  be  made  in  the 
presence  of  NaCl  owing  to  the  opposing  pressure  exerted  by 
the  NaCl,  i.e.  owing  to  the  difference  of  osmotic  pressure  of 
the  NaCl  in  II.  and  in  I.  Assume  for  simplicity's  sake  that  the 
salts  are  all  completely  dissociated  and  equal  volumes  of 
solution  are  present  on  each  side  of  the  membrane.  The 
true  osmotic  pressure  of  the  NaR  is  then  given  by  the  equation 

P0  =  2<iRT 

If  we  call  P  the  opposing  pressure  of  the  NaCl,  we  have  — 
P  =  2(r2  —  *)RT  —  2*RT     or     2(<r2  —  x  —  *)RT 

The  observed  osmotic  pressure  of  the  NaR  in  I.  is  P1} 
where  — 

P!  =  P0  —  P  ==  2RT(«;1  —  (<;2  —  2x))  =  2RT(<:1  —  <T2  +  2x) 
Hence  ^=^±^ 

P0         *i  +  2<Tg 

since  x  =  —  ~  — 


If  ^  is  small  compared  to  ^2»  tnen  PI  —  ^o-  If  ^2  ^s 
small  compared  to  q,  then  Px  =  P0,  as  one  would  expect. 
The  following  table  illustrates  these  relationships. 


OT  0*92 

1  0-67 

2  0'6o 

10  0-52 


3.14       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

On  account  of  the  unequal  distribution  an  addition  of  an 
electrolyte  with  a  common  ion  will  diminish  the  observed 
osmotic  pressure  of  an  electrolytically  dissociated  non-dialysing 
substance.  This  has  been  experimentally  verified  by  Harris. 

The  Case  of  an  Electrolyte  without  an  Ion  in  Common  with  the 
Non-dialysing  Substance. 

This  can  be  treated  in  exactly  the  same  way  as  the  simpler 
case.  Suppose  in  solution  I.  we  have  the  substance  NaR 
(dissociated),  and  in  II.  the  salt  KC1  also  dissociated.  Then 
the  initial  concentration  will  be  represented  by 

Na*     R'     |     K'     Cl' 

*i       c\  c«      ^ 

i.  ii. 

The  ions  of  KCl  diffuse  from  II.  into  I.,  and  the  ions  of 
NaCl  can  diffuse  now  from  I.  into  II.  The  equilibrium  state 
will  thus  be  represented  by— 

Na*      K'      Cl'      R'    |        K'       Na'      Cl' 
c^  —  z     x        y        ci     |     <a  —  x      z      c^—y 

In  order  to  have  electric  neutrality  on  either  side,  it  is 
necessary  that  z  =  x  —  y.  By  considering  the  work  done  in 
small  virtual  changes  at  the  equilibrium  point,  the  following 
relation  is  obtained  as  a  criterion  of  equilibrium. 


=         _ 

[Na']n        K']n      [Cl']i  "        <2 

Taking  the  case  in  which  -1  =  100,  the  following  changes 

cz 
from  the  initial  state  will  take  place  — 

(a)  99  per  cent,  of  the  K*  originally  present  in  II.  will 
diffuse  into  I. 

(b)  Only  i  per  cent,  of  Cl'  originally  present  in  II.  will 
diffuse  into  I. 

(c)  Only  i  per  cent,  of  Na'  originally  present  in  I.  will 
diffuse  into  II. 

It  will  again  be  surprisingly  evident  how  great  an  effect 


the 


"MEMBRANE  HYDROLYSIS"  315 


substance  NaR  has  upon  the  ionic  distribution  of  KC1  on 
the  two  sides  of  the  membrane.  In  this  case  there  has 
apparently  been  an  exceedingly  marked  preferential  effect, 
nearly  all  the  K*  being  "  drawn "  into  I.  and  Cl'  expelled. 
This  latter  phenomenon  would  be  realised  if  the  KG  had 
been  present  in  I.  to  start  with. 

Hydrolytic  Decomposition  of  Salts  by  the  Membrane. 

The  question  which  now  arises  is  :  What  will  happen  if  on 
one  side  of  the  membrane  there  is  NaR  and  on  the  other  pure 
water?  The  Na'  will  tend  to  pass  through  the  membrane 
since  the  latter  is  permeable  to  this  ion,  but  this  can  only  take 
place  if  at  the  same  time  an  equivalent  quantity  of  OH'  (from 
the  water)  diffuses  in  the  same  direction.  The  initial  and 
final  states  could  be  represented  thus — 

Na'     R'     |     pure  water  Na'  Na* 

i.  ii.  H* 

R'  OH' 
i.  ii. 

Initial  state.  Equilibrium  state. 

The  solution  in  compartment  I.  will  thus  become  acid. 
To  find  the  equilibrium  concentration  relations  let  us  assume 
that  equilibrium  is  reached  and  that  a  small  virtual  change  is 
made  involving  the  transfer  of  8n  moles  of  Na*  from  I.  to  II. 
and  8n  moles  of  OH'  from  I.  to  II.  This  leads  to  the 
relation — 


In  order  to  make  the  consideration  of  the  problem  as 
simple  as  possible,  we  assume  the  following  : — 

(a)  Complete  electrolytic  dissociation  of  all  electrolytes 
present  (with  the  exception  of  water  naturally). 

(b}  I.  and  II.  occupy  equal  volumes. 

(c)  The  H'  ions  produced  in  I.  at  the  equilibrium  state  (or 


316       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

the  OH'  produced  in  JI.)  are  in  relatively  great  concentration 
compared  to  the  concentration  of  the  same  ions  produced 
from  water  under  ordinary  circumstances.  We  may  thus 
write  down  the  following  concentration  relations. 

Initial  state.  Equilibrium  state. 

Na'     R'     !     Pure  water        Na'     ET     R'     |     Na'     OH' 

fl         ?!        \  C±  —  X      X        C±  X  X 

I.  II.  I.  II. 

The  equilibrium  relation  above  may  thus  be  written  — 
fi  —  x          x 

x      =[OH']i. 
Also  if  Kw  denotes  the  ionisation  constant  for  water,  then  — 

x  X  [OH']i  =  Kw 
Eliminating  the  [OH']i  term,  one  obtains— 

x*  —  K^ta  —  x) 

If  x  is  small  compared  to  ^  we  obtain  the  very  simple 
relation  — 


This  equation  predicts  very  small  values  for  x,  which 
only  increase  relatively  slowly  with  increase  in  c^,  The 
following  table  shows  the  nature  of  the  results  obtained  for 
the  temperature  25°  C.  (Kw  =  io-14). 


1  00r 

cl 

x 

O'OI 
O'l 

5  x  io~" 

I  X  I0~5 

Q'05% 

0'01% 

I 

2  X  IO-5 

0-002% 

-v 

The  degree  of  hydrolytic  dissociation      brought  about  by 

the  chemically  "  /V/active  "  membrane  is  extremely  small.  The 
remarkable  thing  is  that  there  should  be  any  hydrolysis  at  all, 
especially  as  we  are  considering  the  case  of  complete  dis- 
sociation which  means  that  the  acid  and  base  forming  NaR 
are  both  strong,  as  otherwise  the  H'  present  finally  would 


"MEMBRANE  HYDROLYSIS"  317 

react  with  R'  to  form  undissociated  (weak)  acid.  These 
simple  considerations,  however,  show  that  the  hydrolysis  must 
actually  take  place,  a  fact  which  has  been  experimentally 
verified  by  Harris  in  Donnan's  Laboratory.  By  increasing 
the  volume  of  II.  in  comparison  to  I.  the  degree  of  hydrolysis 
can  be  increased.  If,  for  example,  the  volume  of  II.  is  made 
v  times  greater  than  that  of  I.,  we  must  write  the  equilibrium 
conditions  as  follows  : — 


H'     R' 


Na'     OH' 


x         x 

—  X        X        Ci 

V  V 

I.  II. 

the  corresponding  equations  being — 

x*  =  Kw^2(<r1  —  x) 
or  when  x  is  small  compared  to  q— 

x  = 


If,  for  example,  v  =  100,  ^  =  o'i,  then  x  will  be  of  the 
order  io~4,  so  that  the  percentage  hydrolysis  -        will  be  of 

the  order  OT.  Naturally  if  the  acid  HR  is  a  weak  acid  this 
will  reduce  the  H*  concentration  in  I.  and  will  tend  to  increase 
the  membrane  hydrolysis.  Donnan  has  worked  out  this  case 
when  Ka,  the  dissociation  constant  of  the  acid  HR,  is  small. 
It  is  found  (for  details  the  original  paper  may  be  consulted) 
that  the  degree  of  hydrolysis  may  be  written  — 


If  x  is  small  compared  to  c^  one  obtains  — 


_ .  VJK^V 

~V  K/1 


As  an  example  of  the  numerical  values  likely  to  be  ob- 
tained,  putting  q  =  i,   Kw=  io-14,   and    K«  =  io-5,   then 

x=  io~3,  and  —    —  =  o'i,  i.e.  0*1  per  cent,  hydrolysis.     Such 

€l 
hydrolytic  effects  must  take  place  in  the  ordinary  process  of 


318        A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

dialysis  and  must  alst)  be  present  when  measurements  of 
osmotic  pressure  of  electrolytic  colloids  are  made  by  means 
of  membranes. 


"  Membrane  Potential"  (the  P.D.  existing  when  the  equilibrium 
as  modified  by  the  membrane  is  reached). 

Consider  the  simplest  case  of  NaR  and  NaCl,  equilibrium 
being  represented  by— 

Na- 
R' 


Cl' 


Cl' 


i.  ii. 

Let  77i  and  7r2  be  the  potentials  (for  positive  electricity)  of 
the  solutions  I.  and  II.  Suppose  the  extremely  small  mass 
F8/2  of  positive  electricity  be  transferred  isothermally  from 
II.  to  I.,  then  in  this  virtual  change  of  the  system  from  the 
equilibrium  the  following  work  terms  must  be  considered. 

(a)  Increase  in  free  electrical  energy  =  FS«(TTI  —  7T2). 

(b)  p§n  moles  of  Na'  ions  has  been  transferred  from  II. 
to  I.  and  simultaneously  q&n  moles  of  Cl'  ion  from  I.  to  II., 
where  /  +  q  =  i  (i-c.  p  and  q  represent   the  fraction  of  the 
total  current  carried  by  the  respective  ion,  or  in  other  words 
/  and  q  are  the  transport  numbers  of  the  ions).     The  maximum 
osmotic  work  of  operation  (b)  is  given  by  — 


Now  since  the  system  is  in  equilibrium  the  electrical  virtual 
work  must  balance  the  osmotic  virtual  work,  or  — 

F&fa  -  IT,)  =/3«RT  log  [~-]^  +  ?SnRT  log  -jgt 

Now  we  have  seen  that  in  this  case  in  the  equilibrium  state 
the  following  relation  holds  — 


n-.,^ 
[Na']i  ~'- 
and  also  /  +  q  = 


"MEMBRANE  POTENTIAL"  319 

Hence  it  follows  that — 

RT  . 

7T!  — 772=:-—    log  A 

For  the  simple  case  investigated  we  see  that — 

—  x 


where  

fi  +  2% 

So  that  the  potential  difference  E  across  the  membrane  due 
to  the  distribution  of  ions  is — 


E  =  7T2  —  TTi  =  —  log    i  =  0-058  log  ^ 

E  =  0-058  log  (1  +  -1) 
If  c%  is  small  compared  to  ^  it  follows  that — 
E  =  0-058  log  -1- 

If,  on  the  other  hand,  ^  is  small  compared  to  <r2>  tne 
potential  difference  approximates  to  zero,  as  one  would 
expect,  for  in  the  limit  in  which  there  is  no  NaR  present  at 
all  the  NaCl  will  distribute  itself  in  equal  concentration  on 
each  side  of  the  membrane.  The  following  table  illustrates 
the  numerical  values  of  E  for  a  series  of  arbitrarily  chosen 
c-»  and  ^  values. 


i 
10 

100 
1000 


+  0-017 

+  o'o6o 
+  o'ii6 
+  0-174 


In  a  similar  manner  one  may  calculate  the  potential 
differences  in  the  more  general  case  in  which  KC1  is  present 
in  place  of  NaCl. 


CHAPTER   XI 

Systems  not  in  equilibrium  studied  from  the  thermodynamic  standpoint — 
Affinity  and  its  measurement  by  means  of  vapour  pressure,  solubility, 
and  electromotive  force — Oxidation  and  reduction  processes — Change 
of  affinity  with  temperature. 

CHEMICAL  AFFINITY. 

WE  now  pass  on  to  the  question  of  chemical  affinity  and  its 
measurement.  The  conception  of  some  kind  of  attractive 
force  or  affinity  between  portions  of  matter  is  one  of  the 
oldest  in  science,  but  until  recently  it  never  got  beyond  the 
stage  of  obscure  definition.  It  evaded  quantitative  measure- 
ments, although  attempts  were  made  in  this  direction  by 
Berzelius,  Mitscherlisch,  Wilhelmy,  Guldberg,  and  Waage. 
What  at  first  sight  appears  to  be  the  most  promising  definition, 
namely,  that  the  speed  of  reaction  gives  a  measure  of  the 
affinity,  cannot  be  retained  when  we  remember  how  depen- 
dent reaction  velocity  is  on  a  variety  of  circumstances,  quite 
unconnected  with  the  process  itself,  e.g.  the  presence  of 
catalysts.  The  first  successful  solution  of  the  problem  is 
due  to  Helmholtz,  but  Helmholtz  did  not  pursue  it.  It  was 
really  rediscovered  by  van  't  Hoff  in  1883,  who  came  to  the 

AFFINITY  (Historical  Note). — The  measure  of  affinity,  say,  of  a 
solution  for  the  solvent  being  given  quantitatively  by  the  change  in  free 
energy  involved  in  the  transfer  of  unit  mass  (say  i  mole)  of  solvent  from 
one  to  the  other,  this  being  calculable  for  the  reversible  case,  by  the 

riF       [P 

vapour  pressures  according  to  the  equation  (~-  =  /   yelp,  was  first  given 

Qm     J  p 

by  Helmholtz  (see  two  papers,  Sitzungsberichte  der  Akademie  der  Wissen- 
schaft  zu  Berlin,  1882).  In  his  own  words  "  —  ~  is  to  be  distinguished  as 

G/tfZ 

the  force  with  which  the  water  of  the  solution  is  attracted." 


CHEMICAL  AFFINITY  321 

conclusion  THAT  THE    ONLY  TRUE    MEASURE   OF   CHEMICAL 

AFFINITY  BETWEEN  SUBSTANCES  WHICH  MANIFESTS  ITSELF  BY 
CHEMICAL  REACTION  WHEN  THE  SUBSTANCES  ARE  BROUGHT 
INTO  CONTACT  IS  GIVEN  BY  THE  MAXIMUM  EXTERNAL  WORK 
AT  CONSTANT  TEMPERATURE  AND  AT  CONSTANT  OR  PRACTI- 
CALLY CONSTANT  VOLUME  WHICH  IS  DONE  BY  THE  SYSTEM  IN 
PASSING  FROM  THE  INITIAL  STATE  TO  THE  STATE  FINALLY 
REACHED  BY  THE  REACTION,  i.e.  THE  EQUILIBRIUM  POINT. 

Note,  for  reactions  which  involve  a  volume  change  we  sub- 
tract or  add  the  work  term  involved  by  this  change,  and  what 
remains  is  the  affinity.  To  take  a  simple  physical  illustration 
of  a  "  reaction  "  which  involves  practically  no  volume  change. 
Suppose  two  vessels  containing  aqueous  solutions  of  salt  at 
two  different  concentrations  ^  and  c2  (^  >  c2)  are  brought 
into  contact.  Will  any  diffusion  occur  ?  On  van  't  Hoff's  idea 
we  would  say  yes,  provided  external  work  can  be  done  by  the 
system  in  the  process.  Evidently,  if  we  transport  i  mole  of 
salt  from  the  greater  concentration  to  the  less,  positive  work 
will  be  done  by  the  system  amounting  in  the  most  favourable 
case,  i.e.  as  a  maximum  limit,  to  the  expression — 

RT  log  ^  per  mole  diffused. 

^2 

Hence  we  would  expect  the  change  to  be  a  diffusion  from 
strong  to  weak,  where  ^  >  ^2,  and  such  is,  of  course,  the  actual 
case.  Further,  the  above  expression  shows  that  when  c-^  =  <r2 
there  is  no  nett  work  gained  or  lost  by  transferring  i  mole 
from  one  vessel  to  the  other  at  constant  temperature.  But 
this  is  a  definition  of  the  equilibrium  point,  and  hence  equili- 
brium should  be  reached  when  the  concentrations  are  identical. 
Such  is  actually  the  case.  It  will  thus  be  seen  how,  in  this 
example,  van  't  Hoffs  idea  of  maximum  work  as  a  measure 
of  affinity  or  tendency  to  react  fits  the  facts  well.  It  might  be 
pointed  out  (for  reasons  to  be  given  shortly)  that  practically 
no  heat  effects  take  place  in  the  above  case.  Van  't  Hoff's 
view  as  to  the  real  cause  of  a  reaction  was  not  universally 
accepted  at  first. 

As  long  ago  as   1854  another  view,  and  at  first  sight  a 

T.P.C.— II.  Y 


322        A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

• 
plausible  one,  had  been  put  forward  by  Thomsen,  and  later,  in 

1867,  was  supported  by  Marcelin  Berthelot,  who  considered 
the  heat  effect  as  being  the  determining  factor.  This  is  known 
curiously  as  the  principle  of  maximum  work.  This  principle 
was  a  very  simple  one,  since  it  stated  merely  that  the  heat 
which  is  developed  by  a  chemical  change  indicates  the  direction 
in  which  a  change  will  proceed ;  when  the  possibility  of 
evolution  of  heat  exists,  then  the  reaction  will  proceed  in 
such  a  direction  as  to  bring  this  about.  Take,  for  example, 
hydrogen  and  oxygen  gas.  Two  grams  of  the  former  with 
1 6  grams  of  the  latter  will  develop  69,000  calories,  when 
uniting  to  form  water.  The  principle  just  referred  to  sees  in 
this  heat  development  the  cause  of  the  formation  of  the  water, 
which  as  we  know  takes  place  when  the  mixture  of  the  two 
gases  is  ignited.  Conversely,  if  we  consider  nitrogen  and 
chlorine  we  find  that  by  their  union  no  heat  is  developed ; 
on  the  contrary  heat  is  absorbed.  So  here,  instead  of  the 
union  of  the  elements,  the  tendency  is  towards  the  decom- 
position of  the  compound.  When  the  union  has  been  achieved 
by  indirect  means  the  decomposition  can  be  brought  about 
by  the  slightest  shock.  For  many  years  this  was  a  funda- 
mental principle  of  thermochemistry,  and  many  facts  were 
known  to  support  it.  In  spite  of  this  it  is  not  difficult  to 
furnish  examples  of  cases  in  which  chemical  changes  take 
place  with  the  absorption  of  heat.1  Freezing  mixtures,  like 
that  of  hydrochloric  acid  and  Glauber's  salt,  whose  operation 
depends  on  the  accomplishment  of  a  chemical  reaction  thus — 

Na2SO4ioH2O  +  HCl->  2NaCl  +  ioH2O  +  H2SO4 

really  contradict  the  principle  of  Berthelot.  Further,  the 
majority  of  reactions  proceed  only  to  a  certain  limit,  this 

1  The  lack  of  the  validity  of  the  Thomsen-Berthelot  principle  is  shown 
when  one  applies  thermodynamics  to  chemical  problems.  Horstman, 
1869  (Ostwald's  Klassiker},  was  the  first  to  show  the  way  of  applying  these 
thermodynamical  principles.  A  few  years  later  Lord  Rayleigh  (Proc. 
Roy.  hist.,  7,  386,  1875)  questioned  the  validity  of  the  Thomsen-Berthelot 
principle  in  a  short  paper  "on  the  dissipation  of  energy."  Rayleigh 's 
views  were  further  emphasised  and  extended  to  electrical  systems  by 
Helmholtz  in  1882. 


CHEMICAL  AFFINITY  323 

being  true  of  all  reactions  in  homogeneous  systems  (gas  or 
solution).  Thus  let  us  bring  equivalent  quantities  of  gaseous 
hydrochloric  acid  and  ammonia  into  a  given  space;  a  part 
of  the  gases  will  unite  to  form  solid  ammonium  chloride,  and 
the  production  of  this  salt  will  extend  to  the  point  corre- 
sponding to  its  dissociation  pressure  at  the  given  temperature. 
On  the  other  hand,  let  us  bring  "  solid  "  ammonium  chloride 
into  a  given  space  at  the  same  temperature ;  dissociation  takes 
place,  i.e.  the  substance  which  was  formed  in  the  first  case  is 
decomposed  in  the  second.  But  in  the  first  case  we  are  dealing 
with  an  exothermic  reaction  ;  in  the  second  case  with  an  endo- 
thermic  one.  Berthelot's  principle  postulates  the  existence  of 
exothermic  reactions  only.  In  general,  therefore,  every  single 
instance  of  reversible  reaction  is  sufficient  to  disprove  the 
universal  validity  of  Berthelot's  principle.  Further  instances 
may  be  cited.  Suppose  we  have  ice  and  water  in  contact 
with  one  another  at  o°  C.,  we  know  that  there  is  no  tendency 
for  either  to  increase  at  the  expense  of  the  other,  i.e.  they  are 
in  equilibrium.  The  experimental  fact  that  at  the  equilibrium 
point  their  vapour  pressures  are  identical  is  thus  in  agreement 
with  the  van  't  Hoff  definition  of  affinity.  On  the  other  hand, 
a  very  large  heat  effect,  i.e.  an  evolution  of  80  calories  per 
gram,  occurs  on  solidifying  the  water,  and  if  Berthelot's  principle 
held  good  we  would  naturally  expect  the  system  to  change 
into  the  solid  state  completely.  Also  with  reference  to  the 
diffusion  experiment  in  the  case  of  two  solutions  of  salt  at 
different  concentrations — the  heat  effect  is  immeasurably 
small,  but  the  "reaction"  takes  place  all  the  same.  Here 
again  the  heat  effect  is  no  measure  of  the  affinity.  On  the 
other  hand,  the  vapour  pressures  being  different  (or  the  osmotic 
pressures  which  are  related  to  the  vapour  pressures  thus, 

pQ  —  p  =  -  P  J  would  lead  one  on  van  't  Hoffs  view  to  predict 

a  diffusion  from  the  stronger  to  the  weaker  solution.  The 
accomplishment  of  work  and  the  development  of  heat  in  a 
chemical  change  do  not  mean  therefore  the  same  thing.  They 
often  go  hand  in  hand  as  in  the  case  of  explosives,  like  gun- 
powder and  dynamite.  A  compound  like  phosphonium 


324       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

• 

chloride  (PH4C1)  solid,  however,  tends  to  decompose  at 
ordinary  temperatures  into  gaseous  PH3  and  HC1  with  a 
marked  absorption  of  heat.  Yet  the  decomposition  products 
of  this  compound  may  exercise  a  pressure  of  some  twenty 
atmospheres.  "  Here  we  have  a  case  where  the  possibility 
of  accomplishing  work  does  not  coincide  with  the  capacity  to 
develop  heat,  and  yet  where  it  is  obviously  the  capacity  to  do 
work  which  controls  the  direction  of  the  change  "  (van  't  Hoff). 
The  expression  maximum  work  was,  however,  fortunately 
chosen  by  Berthelot,  since  the  correct  principle  for  the  pre- 
diction of  a  reaction  must  connect  the  possibility  of  the 
change  with  the  possibility  of  a  concomitant  accomplishment 
of  work.  In  order  to  ascertain  the  change  of  free  energy  or 
the  amount  of  external  work  done,  which  is  associated  with  a 
chemical  reaction,  we  must  cause  the  reaction  to  occur  isother- 
mally  and  reversibly ;  and  thereby  we  can  obtain  directly 
the  desired  information  respecting  the  amount  of  maximum 
external  work  which  can  be  obtained  from  the  chemical 
change.  Let  us  suppose  that  under  the  conditions  described, 
the  change  may  occur  in  any  one  of  several  ways ;  even  then 
the  change  in  free  energy  would  always  be  the  same.  For 
otherwise  we  could  complete  the  change  in  one  way,  and  then 
we  could  come  back  by  the  other,  and  thus  we  could  establish 
a  reversible  isothermal  cyclic  process  by  means  of  which  any 
arbitrary  amount  of  external  work  could  be  performed  at  the 
cost  of  the  heat  of  the  environment.  But  this  is  perpetual 
motion,  and  contrary  to  the  Second  Law  of  Thermodynamics, 
and  thus  we  obtain  the  theorem  : 

The  change  of  the  free  energy  of  a  chemical  process  is  inde- 
pendent of  the  way  in  which  the  change  is  completed  as  long  as  it 
is  reversible,  and  is  determined  solely  by  the  initial  and  final 
states  of  the  system. 

We  are  thus  at  liberty  to  make  measurements  of  work  by 
quite  different  methods  applicable  to  different  cases,  and  con- 
sider the  results  as  comparable,  provided  only  that  each  single 
process  be  a  reversible  isothermal  one.  We  have  already  seen 
that  for  practical  purposes  there  are  two  methods  of  measuring 
external  work  which  are  in  frequent  use:  (i)  from  vapour 


CHEMICAL  AFFINITY  325 

pressure  determinations  or  the  equivalent  osmotic  pressure  in 

[l 
the  case  of  solutions,  using  the  three-stage  work  term  /  vdp 

per  gram-mole  transferred,  and  (2)  electromotive  force 
measurements. 

Let  us  take  an  example  of  how  affinity  can  be  measured  by 
means  of  vapour  pressure  data.  Well  known  instances  are  the 
formation  of  hydrated  copper  sulphate,  i.e.  the  affinity  of  copper 
sulphate  for  water;  ferric  chloride  hydration;  sulphuric  acid 
and  water.  Let  us  take  the  latter  case  and  suppose  the  question 
is:  What  is  the  affinity  of  water  for  concentrated  sulphuric  acid 
at  the  temperature  T,  given  that  pQ  is  the  vapour  pressure 
over  pure  water  and  p±  is  the  vapour  pressure  of  water  vapour 
over  concentrated  sulphuric  acid  ? 

The  answer  is  that  the  affinity  "  A  "  of  the  water  for  the 

/Po 

acid  is  /    vdp,  and  if  the  gas  law  is  obeyed  — 
J  PI 


Pi 

One  may  note  in  passing  that  if  we  could  get  absolutely 
anhydrous  sulphuric  acid  /i  —  o  and  log  p±  =  —  oo  ,  or 
A  =  +  co  ,  i.e.  the  affinity  of  the  reaction  would  be  infinitely 
great.  As  a  matter  of  fact,  such  a  state  of  things  is  unrealisable 
experimentally.1 

Again  take  the  case  of  the  substances  represented  by 
CuSO4^H2O.  We  know  that  below  a  certain  temperature 
called  the  transition  point  CuSO4  .  5H2O  is  a  stable  crystalline 
solid.  Above  a  certain  temperature  it  melts,  giving  a  saturated 
aqueous  solution  of  CuSO4  .  sH2O.  We  thus  have  the 
reaction  — 

CuSo  .    HO  ->  CuSO  .    HO  +  2H2O 


Above  the  transition  point  the  reaction  occurs  from  left  to 
right.  There  must  therefore  be  a  positive  tendency  or  affinity 
in  this  direction,  and  hence  in  order  to  make  the  affinity  term 

1  The  idea  that  the  vapour  pressure  of  a  (pure)  substance  is  a  true 
measure  of  the  "active  mass"  of  the  substance  is  due  to  Guldberg  and 
Waage  (1867).  Cf.  A  Pousot,  Comptes  Rendns,  130,  829  (1900). 


326       A   SYSTEM  OF  PHYSICAL    CHEMISTRY 

positive  the  vapour  pressure  of  the  water  over  the  penta- 
hydrate  above  its  transition  point  must  be  greater  than  the 
vapour  pressure  of  the  water  over  the  saturated  trihydrate 
solution,  and  this  has  been  experimentally  verified.  In  general, 
when  dealing  with  transitions  from  unstable  to  stable  phases, 
the  vapour  pressure  of  a  given  compound  over  the  imstable 
phase  is  always  greater  than  its  vapour  pressure  over  the  stable 
phase,  this  as  a  matter  of  fact  being  the  cause  of  the  change. 
Below  the  transition  point,  the  vapour  pressure  of  water  on  the 
saturated  [CuSO4  .  3H2O  -j-  2H2O]  is  greater  than  that  over 
the  crystalline  pentahydrate  and  the  direction  of  chemical 
change  is  reversed.  Note  that  you  must  always  in  such  in- 
stances of  transition  take  into  account  the  whole  system  on 
either  side  of  the  equation.  In  the  simple  case  considered  we 
were  only  dealing  with  the  transfer  of  a  single  substance  (say, 
water)  from  an  initial  to  a  final  state,  the  reaction  being 
carried  out  by  this  transfer.  In  general,  however,  we  have  to 
deal  with  the  simultaneous  reaction  of  two  or  more  substances 
which  give  rise  to  new  substances,  e.g.  2H2  +  O2  =  2H2O.  In 
reactions  which  are  characterised  by  the  existence  of  an 
equilibrium  point  and  an  equilibrium  constant,  i.e.  reactions 
occurring  in  gaseous  mixtures  or  in  solutions,  the  expression 
for  the  work  of  transforming  certain  amounts  of  the  reactants 
(at  a  given  temperature,  pressure,  or  concentration)  into 
resultants  at  the  same  temperature,  but  'at  another  arbitrary 
pressure,  or  concentration,  is  evidently  the  van  't  Hoff 
isotherm,  viz. — 

A  =  RT  log  K  -  RT2k  log  C 

If  we  are  going  to  measure  affinity  by  maximum  work,  we 
must  therefore  regard  A  as  likewise  representing  the  affinity 
of  the  process  considered.  The  formula  shows  directly  that 
when  the  arbitrarily  chosen  concentration  terms  occurring  in 
the  term  2v  log  C  are  identical  with  the  equilibrium  con- 
centration terms,  A  =  o,  i.e.  the  affinity  is  zero  as  one  would 
expect.  In  fact,  the  farther  the  arbitrary  concentration  terms 
are  from  the  equilibrium  values  the  greater  is  the  value  of  A. 
This  expression  brings  out  conversely  the  extreme  importance 


MEASUREMENT  OF  AFFINITY  327 

of  the  equilibrium  constant  in  the  determination  of  affinity 
measurements.  Note  :  An  expression  such  as  "  the  affinity  of 
oxygen  for  hydrogen  "  by  itself  means  nothing  at  all  until  we 
specify  the  concentration  of  the  reactants  and  the  concentra- 
tion of  the  resultants  which  we  are  aiming  at,  and  the 
temperature. 

There  is  one  special  case  to  be  noted.  Suppose  we  start 
with  substances  at  unit  concentration  and  end  with  substances 
also  at  unit  concentration  (i.e.  i  mole  of  each  per  liter,  which 
would  hold  equally  for  gases  or  solutions,  or,  say,  at  i  atmo- 
sphere pressure  of  each,  which  would  apply  to  gases  only), 
then  the  arbitrary  concentration  terms  are  each  unity,  and 
since  the  log.  of  unity  is  zero,  the  expression  for  the  affinity 
becomes — 

A  =  RT  log  K 

The  thermodynamic  significance  of  the  equilibrium  constant 
is  therefore  this :  Its  logarithm  is  proportional  to  the  maxi- 
mum work  involved  in  any  given  reaction  in  which  we  start 
with  the  reactants  at  unit  concentration  and  end  up  with  the 
resultants  also  at  unit  concentration. 


Illustrations  of  the  Measurement  of  Affinity  by  the  Maximum 
Work  produced  by  the  Reaction.  (Cf.  .Sackur's  Chenrische 
Affinitdt  und  ihre  Messung^  Die  Wissenschaft  Series.) 

Let  us  first  consider  an  instance  of  HOMOGENEOUS  GASEOUS 
reactions.  The  affinity  is  given  by  the  expression — 

A  =  RT  log  K  —  RT2?v  log  C 

Affinity  must  be  specified  with  regard  to  the  number  of 
molecules  of  a  given  species  taking  part  in  the  reaction.  If 
we  consider  twice  the  number  of  molecules  the  affinity  is 
doubled,  and  so  on.  Take  as  a  special  case  the  affinity  of 
hydrogen  and  iodine  gases  for  one  another  at  a  given  tempera- 
ture. If  K  is  the  equilibrium  constant  and  [H2]  \\$\  repre- 
sent the  arbitrary  concentration  of  the  two  substances  in  the 
initial  state,  and  [HI]  the  final  concentration  of  the  hydriodic 


328        A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

acid  reached,  then  the*affinity  of  the  process  H2  +  I2  "^ 
per  mole  of  hydrogen  or  iodine  is — 

[HI!2 
A  =  RT  log  K  -  RT  log  ffrrf^ 

LH2j  L12j 

where  ^^C^X^i 

Suppose  the  initial  arbitrarily  chosen  concentration  of 
hydrogen  and  iodine  is  unity,  and  the  concentration  of  the 
hydriodic  acid  is  also  to  reach  unity,  then  the  affinity  of  the 
reaction  under  these  conditions  is  simply— 

A  =  RT  log  K 

This  case  will  now  be  illustrated. 

A  series  of  equilibrium  constants  at  various  temperatures 
have  been  calculated  by  Haber  from  Bodenstein's  data  and  are 
given  in  Chap.  III.,  Part  I.  At  300°  C.  (573°  abs.)  the  value  of 

4/j  _  x)2    * 
K  which  we  require,  namely  -       g        ,  is  (8-9)2  =  80  approx. 

vV- 

The  affinity  per  mole  of  hydrogen  or  iodine — 

80  =  +  5°oo  calories  approx. 


°*4343 

Again  at  the  temperature  500°  C.  (773  abs.)  K=4i 
approx.  Hence  under  the  particular  conditions  of  concentra- 
tion chosen,  the  affinity  is  given  by — 

A  = —  l°gio  41  ==  ~H  ST^0  calories 

°'4343 

It  is  interesting  to  note  that  the  heat  of  the  reaction  is 
negative,  namely  —  6000  calories  approx.  It  is  an  example 
of  where  the  Berthelot  principle  breaks  down. 

Another  interesting  case  is  the  affinity  of  oxygen  for  hydrogen. 
Nernst  and  v.  Wartenberg  have  shown  that  at  1000°  abs.  the 
degree  of  dissociation  of  water  vapour  at  i  atmosphere  pressure 
is  3*0  X  10  \  per  cent.  That  is,  the  fractional  amount  of  i 
molecule  of  water  dissociated  according  to  the  equation 
H2O ->  H2 -j- JO2  is  3-0  X  io~7.  There  are  therefore 
present  in  equilibrium  at  this  temperature  and  pressure 


MEASUREMENT  OF  AFFINITY  329 


3*0  X  io~7  moles  of  hydrogen  and  —  .  io~7  moles  of  oxygen 

for  every  mole  of  water  (neglecting  the  actual  decrease  in 
water  molecules,  since  the  dissociation  is  so  extremely  small). 
For  the  equilibrium  constant  in  terms  of  partial  pressures  (the 
equilibrium  concentration  being  denoted  by  the  suffix  e)  we 
have  therefore — 


i  35  ^>  *^ 

^H20 

This  is  the  dissociation  constant  of  water  vapour  under  the 
given  conditions  of  temperature  and  pressure.  The  reaction 
tve  are  considering  is  just  the  reverse  of  this,  namely, 
2H2  +  O2  ->  2H2O,  and  we  wish  to  find  what  the  affinity  of 
oxygen  for  hydrogen  is,  both  being  at  i  atmosphere  pressure 
(in  separate  vessels)  at  1000°  abs.,  the  reaction  to  take  place 
so  that  the  water  vapour  formed  will  finally  be  at  i  atmo- 
sphere pressure,  also  at  1000°  abs.  The  expression  per  mole 
of  oxygen  is  given  by — 

A  =  RT  log  K'  -  RTZV  log/ 

K'  being  the  reciprocal  of  K.     We  can  thus  write  A  in  the 
form  — 

A  =  RT  log    3  AH2° RT  log  -o 

the  arbitrarily  chosen  values  denoted  by  /Ha,  /()2,  /H20  have 
all  been  taken  to  be  unity  (at  i  atmosphere  each),  so  that  A 

reduces  to  —  RT  log  K  or  —  RT  log  - 

That  is,    A  =  —1-98  X  1000  X  2-3  log  1*35  X  io~20 
=  -}~90}6oo  calories 

This  shows  that  there  is  a  large  positive  affinity  between 
hydrogen  and  oxygen  gases  when  each  is  at  T  atmosphere 
pressure  at  the  temperature  1000°  abs. 

The  affinity  of  processes  occurring  in  solution  can  be 
treated  in  a  similar  way.  An  interesting  case  is  the  affinity 
of  the  process  of  electrolytic  dissociation  in  the  case  of  weak 


330       A    SYSTEM   OF  PHYSICAL   CHEMISTRY 

electrolytes  to  which  the  Mass  Law  in  the  form  of  the  Ostwald 
Dilution  Law  applies.  If  k  is  the  dissociation  constant  of  a 
weak  electrolyte,  then  RT  log  k  represents  the  work  which 
would  be  necessary  to  cause  complete  breaking  up  of  the 
molecules  into  ions  in  a  normal  solution.  Since  k  has 
been  shown  by  experiment  to  be  always  less  than  unity,  it 
follows  that  the  work  is  negative,  i.e.  the  affinity  of  complete 
dissociation  is  negative,  so  that  a  normal  solution  of  free  ions 
is  unstable,  and  combination  to  form  undissociated  mole- 
cules results.  The  "  avidity  "  of  a  weak  acid  for  a  base  we 
have  already  seen  is  put  proportional  to  the  dissociation 
constant  k  of  the  acid  (Ostwald  and  Arrhenius).  From  the 
standpoint  of  the  present  discussion,  it  would  be  more  correct 
to  regard  RT  log  k  as  the  true  measure  of  the  avidity. 

We  can  now  pass  on  to  the  consideration  of  some  typical 
heterogeneous  reactions. 

Formation  of  Salts  containing  Water  of  Crystallisation. 

The  case  which  we  shall  consider  is  the  affinity  of  ice  *  at 
o°  C.  for  various  salts,  some  partially  hydrated,  others  anhydrous 
(cf.  Schottky,  Zeitsch.  physik.  C/iem.,  64,  422,  1908).  One  mole 
of  water  is  vaporised  from  the  ice,  its  pressure  altered  to  that 
in  equilibrium  with  the  salt,  and  then  the  vapour  is  imagined 
to  be  compressed  into  the  salt.  The  process  is  simply  the 
familiar  three-stage  distillation,  the  maximum  work  or  affinity 
being — 

A  =  RT  log^ 
P\ 

Assuming  that  the  vapour  obeys  the  gas  laws,  pQ  is  the  vapour 
pressure  over  ice,  and  p±  is  the  pressure  of  water  vapour  in 
equilibrium  with  the  salt.  It  has  already  been  pointed  out  in 
the  Phase  Rule  chapter,  that  in  the  case  of  dissociating  salts 
such  as  CuSO4-5H2O->CuSO4  3H2O-f-2H2O,  the  dis- 
sociation pressure  of  the  pentahydrate  p±  can  be  taken,  as  far 
as  affinity  is  concerned,  as  likewise  the  equilibrium  vapour 

1  Strictly  one  cannot  speak  of  the  affinity  of  a  substance  but  of  the 
affinity  of  a  process. 


MEASUREMENT  OF  AFFINITY  331 

pressure  p±  of  the  trihydrate,  since  both  these  salts  are  neces- 
sarily present  to  fix  the  equilibrium.     Taking  the  above  case 


as  an  example,  therefore,  the  expression  RT  log      measures 

the  affinity  of  ice  for  the  trihydrate  CuSO4  .  3H2O.  A  series 
of  vapour  pressure  measurements,  by  means  of  the  Bremer 
Frowein  tensimeter  for  example,  are  therefore  sufficient  to 
allow  one  to  calculate  (say,  in  calories)  the  affinity  of  the 
reaction.  At  the  same  time  it  is  interesting  to  compare  the 
heat  evolved  or  absorbed  by  the  same  reaction.  Thomsen 
has  carried  out  a  long  series  of  such  measurements  on  the 
heat  of  the  reaction  between  liquid  water  and  various  salts 
(see  Thomsen's  Thermochemistry  ',  translated  by  Miss  K.  A. 
Burke.  Ramsay  Series).  In  passing  from  ice  to  liquid  water, 
80  calories  per  gram,  or  1440  calories  per  gram-mole  are 
absorbed.  It  is  necessary,  therefore,  to  subtract  1440  calories 
from  Thomsen's  values  to  obtain  the  heat  of  reaction  between 
ice  and  the  corresponding  salt.  This  has  been  done  by 
Schottky,  who  has  compiled  a  table  which  allows  one  to 
compare  directly  the  affinity  of  the  reaction  with  the  heat 
evolved.  Some  of  the  data  of  Schottky's  table  are  reproduced 
below.  It  will  be  observed  that  the  Q  and  A  values  though 
of  the  same  order  of  magnitude  are  by  no  means  identical. 
The  greatest  differences  occur  in  the  case  of  those  substances 
quoted  towards  the  end  of  the  table.  There  is  no  doubt  that 
these  differences  exist  although  it  is  to  be  remembered  at  the 
same  time  that  the  accuracy  of  measurement  of  the  vapour 
pressures  in  some  cases  is  not  very  great  when  the  actual 
pressure  is  small.  It  is  an  interesting,  though  accidental 
relation  that  in  all  the  cases  quoted  except  two  A  is  less 
than  Q. 


332        A   SYSTEM   OF  PHYSICAL   CHEMISTRY 


COLLECTION  OF  VALUES  OF  A  AND  Q  FOR  ICE  AND  ANHYDROUS  OR 
HYDRATED  SALTS. 

(Schottky,  Zeitsch.  physik.  Chem.,  64,  422,  1908). 


Reaction. 

Q  per  mole  of  ice  in 
calories. 

A  per  mole  of  ice  in 
calories. 

Q-A 

CuSO43Aq  +  2Aq  . 

1190 

840 

350 

ZnSO4iAq  +  5Aq   . 

554 

480 

74 

MnSO4i  Aq  4-  4Aq  . 

336 

359 

FeSO46Aq4-  lAq   . 

200 

275 

-75 

Na2S2O3  +  5Aq 

1056 

962 

94 

NaBr4-2Aq       .      . 

656 

585 

71 

Bad,  +  lAq      .      . 

2010 

1410 

1  60 

(Schottky) 

SrCl,2Aq  4-  4Aq     . 

790 

753 

37 

NaC2H3Oo  +  3Aq   . 

1284 

907 

377 

Na0HPO42Aq  4-  5Aq 

640 

424 

216 

Na^HPOjAq  4-  5Aq 

640 

280 

360 

CuSO4iAq  4-  2Aq  . 

1645 

1045 

600 

Na2HPO4  4-  2Aq     . 

1410 

860 

550 

(COOH)2  4-  2Aq      . 

1560 

810 

750 

BaCl2iAq4iAq     . 

2230 

1060 

1170 

ZnSO46Aq  +  I  Aq   . 

1810 

450 

1360 

MgSO46Aq  4-  lAq  . 

2060 

590 

1470 

SrCl,  +  2Aq       .      . 

2930 

H30 

1500 

CuSO4  +  lAq     .      . 

4860 

2510 

(Difficult  to 

MnSO4  +  lAq    .      .      . 

3990 

measure 
owing  to 

<286o 

small  /. 

ZnSO4+iAq    .      .      . 

6880 

2350 

4530 

The  Affinity  of  Carbon  Dioxide  (CO^for  Lime  (CaO}. 

The  reaction  is  CaO  +  CO2  ->  CaCO3. 

Suppose  we  have  a  large  reservoir  of  carbon  dioxide  at  a 
pressure  /0  at  a  given  temperature.  Suppose  that  the  dissocia- 
tion pressure  of  calcium  carbonate,  i.e.  the  pressure  of  carbon 
dioxide  in  equilibrium  with  lime  and  calcium  carbonate  at 
the  same  temperature,  is  /x,  the  affinity  of  the  reaction  per 
mole  is — 


The  expression  A  =  RT  log  —  is,  of  course,  only  a  special  case  of  the 


MEASUREMENT  OF  AFFINITY  333 

assuming  that  the  carbon  dioxide  obeys  the  gas  law.  If 
the  gas  reservoir  is  at  atmospheric  pressure  /0  =  i,  and 
A  =  —  RT  log/!-  This  is  the  affinity  of  carbon  dioxide  at 
atmospheric  pressure,  and  at  temperature  T,  for  lime  at  the 
same  temperature.  Since  p±  is  less  than  unity  up  to  fairly 
high  temperatures,  this  expression  for  A  is  a  positive  quantity. 
That  is,  if  carbon  dioxide  at  one  atmosphere,  and  at  ordinary 
temperatures,  be  brought  into  contact  with  lime,  the  affinity 
is  positive,  and  the  reaction  resulting  in  the  formation  of 
calcium  carbonate  will  proceed.  If  the  temperature  and 
pressure  values  are  such  that  A  is  negative,  this  means  that 
the  affinity  is  negative,  and  therefore  the  reverse  action  will 
take  place,  namely,  the  dissociation  of  the  carbonate. 

Affinity  of  Oxygen  for  Metals. 

One  of  the  most  important  cases  in  which  one  wishes  to 
measure  the  affinity  is  that  of  the  oxidation  of  metals  by  oxygen 
gas.  In  general  the  reaction  — 


does   not  go   completely  but  reaches   an  equilibrium.     The 

van  't  Hoff  isotherm.     The  reaction  considered  is  C(X  +  CaO  —  >  CaCO3. 
The  equilibrium  constant  K  at  temperature  T 


__ 

CeC02  X  CeCaO 

the  term  C6  denoting  the  equilibrium  concentration  of  the  given  substance 
in  the  vapour  state  (in  contact  with  the  solids).  The  term  Cfc^Q  corresponds 
to  the  partial  pressure  term  p^.  We  wish  to  find  what  the  affinity  of  the 
reaction  is  (per  mole  of  CO2)  when  carbon  dioxide  at  a  pressure  p0  (corre- 
sponding to  a  concentration  C0)  reacts  with  lime,  the  end  products  being 
in  equilibrium,  that  is  possessing  the  values  CeCaCo3>  CeC02>  CtCa0.  The 

xi 

term  RT2f  log  C  is  thus  RT  log  =  --  e--°?  —  ,  and  hence 

CeCaO  X  C0C0 


A  =  RT  log        C^C03-_  _  RT  C*CaC03        =  RT 

CeCaO  x  CeC02  CeCaO 

=  RTlog|-'. 
The  gas  law  is  assumed  throughout. 


334       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

• 

metallic  oxide  possesses  a  certain  decomposition  pressure  at 
each  given  temperature,  but  this  is  small  and  extremely 
difficult  to  measure.  The  affinity  of  metal  for  oxygen  of  any 
given  pressure  can  be  determined  by  reducing  the  oxide  by 
means  of  a  reducing  agent  whose  affinity  for  oxygen  is  known. 
Thus,  if  we  reduce  the  oxide  by  means  of  carbon — 

MeO  +  C  ->  Me  +  CO    .     .     .     .  (i) 
and  then  suppose  this  split  up  into — 

(2) 

(3) 

By  measuring  the  affinity  in  (i)  and  (3)  we  can  calculate  the 
affinity  of  (2)  by  subtraction.  What  we  want  is  the  reverse  of 
equation  (2).  Let  us  take  reaction  (i), 
and  suppose  that  at  a  temperature  T  the 
equilibrium  is  reached  when  CO  has  a 
pressure  /x.  The  first  question  is,  will 
CO,  at  a  pressure  of  i  atmosphere,  when 
brought  into  contact  with  Me,  C,  MeO, 
at  the  temperature  T,  cause  a  reduction 
or  oxidation  of  the  metal  ?  Consider  the 
vessel  in  Fig.  88.  We  can  tell  at  once 
88  by  measuring  the  work  done  in  bringing 

i  mole  of  CO  from  pressure   i  to  pres- 
sure /x.     The  work  Als  i.e.  the   affinity  of  CO  for  Me,  is 

RT  log  —  =  —  RT  log/!.     This  is  the  reverse  of  equation  (i). 
.-.  —  A!  =  affinity  of  [MeO  +  C] 

Now  if  /!  is  less  than  i  atmosphere,  Ax  will  be  positive. 
We  know  from  experience  that  if  we  bring  in  CO  at  i  atmo- 
sphere pressure  there  is  too  great  a  concentration  of  the  gas 
for  equilibrium  to  be  reached,  and  in  accordance  with  the  law 
of  mass  action  some  of  the  carbon  monoxide  must  disappear 
so  as  to  reduce  its  pressure.  It  disappears  by  reacting  with 
the  metal,  giving  oxide  and  carbon,  that  is  to  say  an  oxidation 
takes  place.  If,  however,  p±  were  greater  than  i  atmosphere, 
then,  if  we  were  to  change  the  gas  pressure  from  p±  to  i 


MEASUREMENT  OF  AFFINITY  335 

atmosphere,  we  would  find  that  the  system  would  tend  to 
produce  more  carbon  monoxide  to  bring  its  pressure  up  to  the 
equilibrium  pressure  p±.  That  is  to  say,  a  reduction  of  the 
metallic  oxide  would  take  place.  This  brings  out  clearly  how 
very  careful  one  must  be  in  saying  that  such  and  such  a 
substance  is  a  reducing  agent.  Whether  it  is  a  reducer  or 
not  depends  on  the  conditions  of  the  experiment.  The  above 
expression,  that  is  —  Al5  however,  gives  quantitatively  the 
affinity  of  the  reaction  — 

MeO  +  C  ->  Me  +  CO 

If  we  call  A2  the  affinity  per  mole  of  oxygen  of  the  second 
reaction  as  written,  then  —  A2  is  what  we  want  to  measure, 
corresponding  to  the  reaction  — 

2Me       O 


(The  affinity  reckoned  per  J  mole  of  oxygen  is  JA2.)  Further, 
putting  A3  as  the  affinity  of  reaction  (3)  per  mole  of  oxygen, 
it  follows  that  — 


[If  this  equality  of  work  terms  did  not  hold  good,  then  we 
could  imagine  a  cyclic  isothermal  process,  which  would  yield  a 
continuous  quantity  of  work,  but  this  in  contradiction  to  the 
Second  Law.]  The  quantity  which  we  wish  to  measure, 
namely,  —  A2  is  given  by  — 

—  A2  =  +  2Ai  +  A3 

First  of  all  we  have  to  measure  £A3,  i.e.  the  affinity  of  the 
reaction  — 

£O2  +  C  ->  CO 

[A3  would  be  the  affinity  per  mole  si  oxygen  O2]. 

The  dissociation  of  carbon  monoxide,  however,  even  at 
extremely  high  temperatures  is  immeasurably  small  ;  and  we 
have  thus  to  measure  A3  indirectly  —  by  combining  two  other 
reactions  whose  individual  affinities  can  be  calculated.  Let 
us  take  the  reactions  — 

C02  =  CO-j-i02  .....     (4) 
and  2CO  =  C  +  C02      .....     (5) 


336        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

• 
by  adding  (4)  and  (5)  we  get  — 

CO  =  C  +  J02, 

which  is  equation  (3)  reversed.  Hence  denoting  by  A4  the 
affinity  of  reaction  (4)  per  mole  of  oxygen  (O2),  and  A6  the 
affinity  of  reaction  (5)  as  it  stands,  we  get  — 


Nernst  and  Wartenberg  (Zdtsch.  physik.  Chem.,  56,  548,  1906) 
have  determined  the  dissociation  of  carbon  dioxide  into 
carbon  monoxide  and  oxygen  at  temperatures  from  1300°  to 
1500°  absolute,  and  from  the  data  have  calculated  the  equi- 
librium for  all  other  temperatures.  The  equilibria  relations  of 
equation  (5)  have  been  determined  by  Boudouard  (Comptes 
Rendus,  128,  842,  1899),  so  that  the  affinity  A3  is  known. 
Now  we  know  Ax  also,  and  hence  —  A2  can  be  calculated. 

Example.  —  WHAT  is  THE  AFFINITY  OF  OXYGEN  FOR  IRON 
AT  ATMOSPHERIC  PRESSURE  AND  AT  A  TEMPERATURE  OF 
1000°  ABS.  ? 

According  to  Schenk,  Semiller  and  Falcke  (Ber.^  40,  1708, 
1906),  the  equilibrium  pressure  of  carbon  monoxide  for  the 
reaction— 

FeO  +  C  ->  Fe  +  CO 
is  — 


T 

Aco 

log  /i 

556°  C. 
596-  C. 
666°  C. 

73-2  mm. 
164*2  mm. 
386  'o  mm. 

1-866 
2-215 
2-566 

From  these  numbers  one  obtains  by  exterpolation  that  at 
1000°  absolute,  the  equilibrium  pressure  of  carbon  monoxide 
is  810  m.m.  =  ro6  atmospheres.  Evidently  by  putting  in 
carbon  monoxide  at  i  atmosphere  over  the  solids  more 
carbon  monoxide  will  tend  to  be  formed,  i.e.  reduction  of 
oxide  takes  place.  The  affinity  of  the  reaction,  i.e.  the  work 
obtained  by  transferring  i  mole  of  carbon  monoxide  from 


MEASUREMENT  OF  AFFINITY  337 

a  pressure  of  i  atmosphere  to  a  pressure  of  ro6  atmo- 
spheres is — 

A!  =  RT  log J-  =  -RT  logA 

=  —1-98  X  2-3  X  1000  X  0-025  =  — 114  calories 
/.  -\-  2  A!  =  —  228  calories 

The  dissociation  equilibrium  of  carbon  dioxide  may  be 
obtained  from  Nernst's  data  (Lehrbuch^  5  Aufl.,  p.  680).  At 
1000°  absolute  carbon  dioxide  at  atmospheric  pressure  is  dis- 
sociated to  the  extent  of  1*58  X  io~5  per  cent.,  that  is  i  mole 
of  carbon  dioxide  yields  a  fraction  1-58  X  io~7  mole  of 
carbon  monoxide  and  1*58  X  io~7  atoms  of  oxygen,  or 

— j-  X  io~7  mole  of  O2.     Hence  the  affinity   with  which  i 

mole  of  carbon  dioxide  at  atmospheric  pressure  would  break 
up  into  i  mole  of  carbon  monoxide  and  £  mole  of  oxygen, 
also  at  atmospheric  pressure,  is  at  iooo°- 

iA4  =  RT  log  K  -  Ev  log  C 

2/V  log  C  vanishes  because  all  the  concentrations  are  unity  as 
has  been  arbitrarily  chosen,  i.e.  we  have  made  use  of  the  term 
i  atmosphere.  K  must  of  course  be  calculated  in  pressure 
terms.  Hence — 

=  RT  log  K  =  RT 


•      -  X  io- 

=  2'3  X  i  "98  X  1000  X  log 


i 

=  2-3  X  1-98  X  1000  X  (—10-35) 
=  — 47,200  calories 

The  equilibrium  pressure  of  the  reaction — 


:) 


is,  according  to  Boudouard,  i  atmosphere  at  1000°  abs.,  the 
composition  of  the  gas  mixture  being  67  volume  per  cent,  of 
carbon  monoxide,  and  33  per  cent,  of  carbon  dioxide.  In 
other  words  the  pressure  exerted  by  the  carbon  monoxide  is 
T.P.C. — ii.  z 


338        A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

• 

0-67  atmosphere  and  the  pressure  of  the  carbon  dioxide  0-33 
atmospheres,  the  pressure  of  the  carbon  being  of  course 
extremely  small.  The  equilibrium  constant  of  the  reaction  at 
this  temperature  is  — 


(0-67)2 

The  work  required  to  bring  the  system  consisting  of  carbon 
monoxide  and  carbon  dioxide  each  at  i  atmosphere  pressure 
and  in  the  presence  of  carbon  (this  system  not  being  in  equi- 
librium) into  equilibrium  is  — 

A5  =  RT  log  K  —  RTZV  log  C 

=  RTlog  ^-RT  log  ^|^=-RT  log  1-36 

=  —  2-3  X  1*98  X  1000  X  0-134  =  —  610  calories 

[That  is,  there  is  a  negative  tendency  for  carbon  monoxide 
at  i  atmosphere  pressure  in  contact  with  carbon  and  also  in 
the  presence  of  carbon  dioxide  at  i  atmosphere  to  dissociate 
into  carbon  and  carbon  dioxide,  in  other  words  the  reverse 
change  (i.e.  formation  of  carbon  monoxide)  will  take  place.] 

The  affinity  of  carbon  for  oxygen  (at  i  atmosphere 
pressure),  i.e,  equation  (3)  C  -f  JO2  ->  CO,  forming  carbon 
monoxide  atm  partial  pressure  of  i  atmosphere,  is  |A3  where 

JA3  =  —  JA4  —  A5  =  -J-  47,200  -\-  6ro  =  47,810  calories 

Finally  the  quantity  we  are  aiming  at,  the  affinity  of  oxygen 
(at  i  atmosphere  pressure)  for  iron  at  1000°  abs.  is  given  by 
(  —  A2),  where  — 

—  A2=  +  2AX  +  A3 

=  —228  -f  95,620 

—  ~h  95>392>  which  in  round  numbers  =  95,400 
The  positive  sign  means  that  under  the  conditions  of  the 
experiment  (1000°  abs.  and  i  atmosphere  oxygen  pressure) 
there  is  positive  affinity  between  oxygen  and  iron,  i.e.  iron 
oxide  will  be  formed.  Note  also  that  under  the  same  con- 
ditions the  affinity  of  oxygen  per  mole  for  carbon,  i.e.  -f-  A3 
is  +  95,620,  almost  identical  (numerically)  with  the  affinity  of 
oxygen  for  iron,  being  just  a  little  greater. 


MEASUREMENT  OF  AFFINITY  339 

Finally  it  may  be  pointed  out  that  while  by  the  union  of 
2  moles,  i.e.  gram-atoms,  of  iron  to  i  mole  of  oxygen  at 
i  atmosphere,  the  free  energy  change  is  an  evolution  of 
95,400  calories,  the  heat  of  reaction  as  determined  by  Le 
Chatelier  is  129,900  calories,  that  is  to  say  considerably 
higher. 

Another  method  of  obtaining  the  affinity  of  oxygen  for 
iron  may  be  employed.  The  idea  is  the  same  as  before 
except  that  instead  of  carbon,  one  uses  hydrogen  as  the 
reducing  agent,  that  is  provided  the  affinity  of  hydrogen  for 
oxygen  is  greater  than  the  affinity  of  oxygen  for  the  metal. 
(For  details  see  Sackur,  I c.,  pp.  64  and  65.) 

Electrical  Method  of  Measuring  Affinity. 

Having  considered  a  few  cases  of  affinity  in  which  the 
determination  is  carried  out  by  means  of  vapour  pressure  or 
concentration  measurements,  it  is  necessary  to  discuss  the 
electrical  method,  especially  as  this  may  be  employed  in  cases 
to  which  other  methods  are  quite  inapplicable.  As  already 
pointed  out,  if  we  take  the  faraday  (96,540  coulombs)  as  the 
unit  of  electrical  quantity,  i.e.  the  quantity  of  electricity 
associated  with  i  gram  equivalent  of  any  ion,  and  if  the 
valency  of  the  ion  of  the  chosen  substance  be  ;/,  the  electrical 
energy  connected  with  the  transformation  of  i  gram  ion  is 
n  X  E  or  ;/E,  where  E  is  the  electromotive  force  of  the  cell  in 
which  the  reaction  is  proceeding.  The  reaction  in  the  cell 
must  of  course  be  a  reversible  one,  as  already  explained  in  the 
chapter  on  the  thermodynamic  criteria  of  equilibria.  The 
most  familiar  instance  is  that  of  the  Daniell  Cell,  in  which  the 
following  reaction  takes  place — 

Znmetai  +  Cu++  ->  Zn++  +  Culnetai 


The  ions  in  this  case  are  divalent  (;/  =  2).  If  E  =  e.m.f. 
of  the  cell,  then  the  affinity  (A)  of  the  process  in  terms  of  i 
gram  ion  (the  same  as  i  gram-mole  in  this  case)  of  either 
copper  or  zinc  is  2E.  When  the  reaction  reaches  an  equi- 
librium the  e.m.f.  falls  to  zero,  that  is  the  affinity  is  zero  as 


340        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

• 

one  would  expect.  If  a  Daniell  cell  bo  set  up  with  the  zinc 
sulphate  or  copper  sulphate  solutions  of  the  same  strength 

(  say  —  J,  the  e.m.f.  is  approximately  n  volts.     The  affinity  A 

of  the  process  is  therefore  2*2  volt  faradays.  This  can  very 
easily  be  converted  into  other  units.  Thus  i  faraday 
=  96,540  coulombs,  and  hence  A  =212,  390  volt-coulombs 
or  joules.  Further,  4*2  joules  =  i  calorie,  and  hence 
A  =  50,570  calories  (per  gram-ion  or  per  gram  atom  of  (say) 
zinc).  The  reaction  which  we  are  actually  considering  is  the 
affinity  of  metallic  zinc  for  a  certain  solution  of  copper  sul- 
phate. We  might  also  regard  it  as  measuring  the  relative 
affinities  of  the  metals  for  electricity,  i.e.  their  relative  tendency 
to  pass  into  the  ionic  state.  The  reaction  reaches  an  equi- 
librium which  may  be  denoted  in  the  usual  way  — 


and  the  equilibrium  constant  is  given  by  — 

CeZn++  X  Ccu 
Cecu++  X  Czn 

but  since  the  "  concentration  "  of  the  solid  metal  itself  is  neces- 
sarily a  constant,  we  can  simply  write  — 


If  we  have  copper  sulphate  and  zinc  sulphate  solutions  in 
the  cell  at  arbitrarily  chosen  concentrations,  such  that  Czn++ 
and  Cou++  denote  respectively  the  metallic  ion  concentrations, 
then  the  affinity  of  the  process  at  temperature  T  must  be  given 
by  the  van  't  Hoff  isotherm,  viz.— 

A  =  2E  =  RT  log  K  —  RTZV  log  C 


Now  suppose  we  choose  the  zinc  sulphate  and  copper 
sulphate  solutions  at  approximately  the  same  strength  (say 
decinormal),  this  will  mean  that  the  "  arbitrary  "  concentrations 
of  the  ions  are  approximately  the  same,  and  therefore  the  second 


MEASUREMENT  OF  AFFINITY  341 

term  vanishes.  The  e.m.f.  in  such  a  case  at  room  tempera- 
ture (T  =  298)  is  found  by  measurement  to  be  approximately 
i  'i  volt,  so  that  — 


A  =  2  x  i'i  =  RT  log       "     =  RT  log  K 

WCu++ 

The  electrical  method  of  measuring  A  is  thus  a  very  con- 
venient way  of  obtaining  the  equilibrium  constant.  Working 
out  the  above  example,  we  see  that  — 


Since  2-2  is  expressed  in  volt-faradays,  it  is  necessary  to  have 
RT  also  in  the  same  units.     This  is  done  by  putting  — 

R  =  o'86  X  io-4. 

Hence  log  K  -  ^-^—^  =  34 

or  K  =  io3± 

This  means  that  when  the  Zn++  and  Cu++  have  reached 
equilibrium,  the  concentration  of  Zn++  is  io34  times  that  of 
Cu++.  It  is  evident  therefore  that  in  the  ordinary  precipita- 
tion of  metallic  copper  from  solution  by  metallic  zinc,  the 
solution  will  have  become,  to  all  intents  and  purposes,  pure 
zinc  sulphate,  the  copper  being  quantitatively  precipitated. 
The  equilibrium  point  in  this  case  is  so  far  shifted  over  to  one 
side  that  it  would  be  utterly  hopeless  to  attempt  to  determine  K 
directly  by  analysis.  This  is  a  striking  illustration  of  the  use- 
fulness of  the  electromotive  force  method. 

It  is  of  interest  to  look  at  the  affinity  of  the  process  occur- 
ring in  the  Daniell  cell  from  the  standpoint  of  the  solution 
pressures  of  the  metals  zinc  and  copper.  Suppose  that  a  cell 
having  the  zinc  sulphate  and  copper  sulphate  solutions  at  the 
same  ionic  concentrations  (say  normal)  yields  a  total  e.m.f.  of 
E  volts.  Experiment  shows  that  the  copper  is  the  positive 
electrode,  and  therefore  current  flows  inside  the  cell  from  zinc 
to  copper,  and  outside  from  copper  to  zinc.  Let  us  neglect 
the  P.D.  at  the  contact  of  the  solutions.  The  total  e.m.f.  E  is 
then  due  to  the  two  single  P.P.'s  at  the  zinc  and  capper 


342        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

• 

electrodes  respectively,  viz.  7TZmc  —  ^copper-  Suppose  a  small 
virtual  change  in  the  system  to  take  place,  involving  the 
transfer  of  8F  faradays  through  the  cell  at  constant  tempera- 
ture and  volume.  This  current  is  imagined  to  pass  from  the 
zinc  to  the  copper  through  the  cell.  We  have  to  consider  all 
the  work  terms,  electrical  and  osmotic.  Since  the  charge  trans- 
ferred is  positive  the  electrical  work  is  —  7TZinc8F  -j-  7rCOpPerSF, 
since  the  potential  at  the  zinc  electrode  rises  as  we  pass  to 
solution,  and  we  reckon  work  terms  positive  when  the  force 
assists  the  motion,  negative  when  the  force  opposes  the  motion 
(as  is  the  case  at  the  zinc  electrode).  The  osmotic  work  done 
in  the  sense  of  Nernst's  theory  is  first  the  work  term  — 

5;^RT  log  — 

/Zn+  + 

(where  /zn++  is  the  osmotic  pressure  of  the  zinc  ions,  PZU  the 
solution  pressure  of  the  electrode,  and  8m  is  the  mass  in  moles 

8F 
of  zinc  corresponding  to  a  charge  8F,  i.e.  ^—  =  the  valency 

=  n),  this  being  the  maximum  work  done  in  transporting 
8m  equivalents  of  the  metal  zinc  from  the  zinc  electrode  to 
the  solution.  Now  transport  8m  moles  of  copper  from  the 
copper  solution  to  the  copper  electrode.  The  maximum  work 
is  again  — 


The  total  "  osmotic  work  "  is  the  algebraic  sum  of  these 
quantities,  namely  — 

log  ^  +  &»RT  log^ 

/Zn+  +  PCU 

The  total  virtual  work  done  we  can  equate  to  zero,  is  — 

&»RT  log  —  +  8«RT  iog^-+  —  TTznSF  +  wcuSF  =  o 
#Bn+*  PCU 

_       RT         PZn     ,RT       /cu++ 
E==^F10^^+«F1°§T^ 

RT.         PZn         RT.        /Zn+  + 

E=lo-10 


MEASUREMENT  OF  AFFINITY  343 

But  we  have  assumed  that  we  are  dealing  with  a  case  in 
which  the  concentrations,  and  therefore  the  osmotic  pressures, 
of  the  zinc  and  copper  ions  are  identical.  The  last  term 
therefore  vanishes,  and  we  are  left  with — 


where  the  terms  Czn  and  Ccu  bear  to  Pzn  and  Pen  the  same 
relation  as  concentration  of  ions  in  solution  bears  to  the 
respective  osmotic  pressures.  But  we  have  seen  that  the 
affinity  A=2EF,  or  2E  if  F  be  taken  as  unit  of  current 
(charge). 

2RT  ,         Czn 

Hence  A  —  --  log  - 

n        to  CCu 

and  since  ;/,  the  valency  of  the  ions,  is  2,  this  expression 
becomes  — 


Further,  we  have  seen  that  when  the  ionic  concentration  of 
the  metallic  ions  is  the  same,  the  van  't  Hoff  isotherm  likewise 
reduces  to  the  form  — 

A  =  RT  log  K 

where  K  is  the  equilibrium  constant  of  the  reaction,  i.e.  — 


Hence  K  = 

PCu 

That  is,  the  equilibrium  constant  of  the  reaction  is  identical 
with  the  ratio  of  the  solution  pressures  of  the  metals.  Since 
we  have  shown  that  K  is  of  the  order  io34,  this  extremely 
large  number  likewise  represents  the  number  of  times  the 
solution  pressure  of  zinc  is  greater  than  the  solution  pressure 
of  copper. 

The  greater  the  solution  pressure  P  of  a  metal,  the  more 
completely  will  it  throw  out  of  solution  the  ions  of  a  metal 
having  a  lower  solution  pressure.  The  precipitation  will  tend 
to  become  less  complete  the  nearer  the  solution  pressure 


344        A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

• 

values  are  to  one  another.  Since  the  values  for  P  for  various 
metals  follow  the  order  of  "electrolytic  potential  series,"  it 
is  clear  that  two  metals  in  close  proximity  in  this  series  can 
only  give  rise  to  the  phenomenon  of  incomplete  precipitation 
when  the  one  with  the  higher  value  of  P  is  added  to  a  solution 
of  the  ions  of  the  one  with  a  somewhat  smaller  P  value. 

In  the  case  of  the  Daniell  cell  we  have  really  assumed 
that  the  affinity  is  measurable  in  terms  of  the  electromotive 
force.  The  validity  of  the  electrical  method  (by  comparison 
with  other  results  obtained  by  different  methods)  has  been 
carefully  tested  by  Kniipffer  and  Bredig  (Zeitsch.  physik.  Chem., 
26,  255)  in  the  case  of  the  reaction  between  thallium  chloride 
and  potassium  sulphocyanide  — 

T1.C1     +     KCNS     =     T1CNS     +     KC1 

Sparingly  soluble  Sparingly  soluble 

which  reaches  an  equilibrium  point  determinable  by  analysis. 
The  equilibrium  exists  not  only  at  a  definite  temperature,  as 
in  the  case  of  equilibria  having  transition  points,  but  as  the 
temperature  changes,  is  displaced  gradually  in  one  direction  or 
other  with  a  corresponding  alteration  in  the  concentrations  of 
the  dissolved  potassium  chloride  and  potassium  sulphocyanide. 
The  above  change  was  employed  for  the  construction  of  a 
galvanic  cell  (Kniipffer  and  Bredig,  loc.  cit.}. 

_,    ...         I  KCNS  solution      KC1  solution  I  rp,    ... 
Thalhum  saturated  saturated         rhalhum 

Amalgam   ,      with  T1CNg          with  T1Q     J  Amalgam 

whose  e.m.f.  was  measured.  Assuming  that  the  e.m.f  was  a 
correct  measure  of  the  affinity  (in  this  case,  since  we  are  dealing 
with  monovalent  ions,  the  e.m.f.  should  be  identical  numeri- 
cally with  the  affinity),  the  equilibrium  constants  for  the  tem- 
peratures of  39*9°  C.,  20°  C.,  o'8°  C.,  were  separately  calculated 
from  the  van  't  Hoff  isotherm  — 


E  =  A  =  RT  log  K  —  RT£v  log  C 
and  were  found  to  be  respectively  — 

o-88  1*26  1-79 


MEASUREMENT  OF  AFFINITY  345 

while  purely  analytical  methods  gave — 

0-85  1-24  1-74 

This  agreement  is  very  strong  evidence  for  the  accuracy 
of  the  assumption  which  is  made  above,  that  the  electromotive 
force  is  a  measure  of  the  affinity. 

Further,  since  K  is  a  function  of  the  temperature,  then 
on  working  with  constant  concentrations  and  altering  the  tem- 
perature, i.e.  keeping  the  ZV  log  C  the  same,  it  is  possible  to 
imagine  the  case  in  which  the  first  term  may  equal  the  second, 
and  if  such  a  point  was  reached  A  would  equal  o,  and  if  the 
electromotive  force  was  a  true  measure  of  the  affinity  it  should 
also  be  zero  at  this  temperature.  KntiprTer  found,  experimen- 
tally, using  the  cell  described  above,  that  the  electromotive 
force  was  zero  at  42-3°  C.,  while  the  temperature  calculated 
from  the  isotherm,  at  which  by  starting  with  given  arbitrary 
concentrations,  and  altering  the  temperature  until  these  con- 
centrations coincided  with  the  equilibrium  concentrations,  was 
calculated  to  be  41*3°  C.  This  is  very  strong  proof  of  the 
validity  of  the  principle  involved. 

Electrometric  measurements  may  likewise  be  employed  to 
calculate  the  affinity  of  complex  ion  formation  (e.g.  silver 
cyanide  complex).  For  details  cf.  Sackur,  loc.  tit.,  p.  85,  seq. 

Gas  Cells. 

The  affinity  of  hydrogen  ion  (H*)  for  hydroxyl  ion  (OH'), 
and  the  determination  of  the  ionisation  constant  of  water  from 
electromotive  force  measurements  will  first  be  considered. 

Take  the  following  cell — 

Inside 


Normal 
HC1 


i  Normal 
NaOH 


H 


A  considerable  P.D.  exists  at  the  contact  of  the  acid  and 
the  alkali  (Nernst,  Zeitsch.physik.  Chew.,  14,  155, 1894).  When 
this  is  corrected  for,  however,  the  e.m.f.  of  this  cell  is  0*8 1 
volt,  the  hydrogen  electrode  dipping  in  the  acid  being  the 
positive  pole,  i.e.  current  flows  as  indicated  from  alkali  to  acid, 


346        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

inside  the  cell.  The  pressure  of  the  hydrogen  gas  is  supposed 
to  be  the  same  in  both  cases.  The  chemical  reaction  which  is 
taking  place,  and  whose  affinity  we  are  measuring  is — 

H-  -f  OH'->H2O 

Thus,  from  the  right  hand  electrode  a  gram  ion  of  hydrogen 
goes  into  solution  leaving  the  electrode  negatively  charged.  In 
the  presence  of  normal  NaOH,  i.e.  approximately  normal  OH', 
the  H*  ion  cannot  exist  in  the  free  state  owing  to  the  small 
dissociation  constant  of  water,  and  therefore  a  great  quantity 
of  OH'  is  used  up  to  form  H2O.  The  current  travelling, 
however,  from  right  to  left  is  carried  by  Na*  ions,  and  H* 
with  it,  and  Cl'  and  (OH')  diffusing  against  it.  At  the  left 
hand  electrode  H'  loses  its  charge  and  goes  off  as  gas;  the 
total  result  is  that  hydrogen  gas  is  used  up  at  the  right  hand 
electrode,  an  equivalent  amount  going  off  as  gas  at  the  left 
hand  electrode,  whilst  in  the  cell  itself  a  gradual  neutralisation 
of  the  alkali  by  the  acid  goes  on.  This  neutralisation  is  caused 
by  the  transport  of  electricity  by  Na'  ions  in  one  direction, 
and  Cl'  ions  in  the  opposite,  and  is  to  be  distinguished  from 
the  natural  diffusion  which  would  of  course  go  on  apart  from 
the  e.m.f.  production.  Finally  the  whole  system  becomes  a 
solution  of  sodium  chloride  of  the  same  strength  throughout, 
and  when  this  stage  is  reached  the  e.m.f.  is  zero,  the  two 
electrodes  being  identical  and  acting  in  opposite  directions. 
Considering  the  condition  of  things  whilst  the  e.m.f.  is  still 
constant  (at  the  value  o'8i  volt)  the  neutralisation  process 
is  confined  to  the  surface  of  contact,  the  affinity  A  can  be 
written — 

A  =  RT  log,  K!  —  RT  loge 


X  COHJ 


Here    Ki  =  -  —  ^°  —  .      The    ionisation   constant    of 

'   X    CtOH' 


water,  which  is  denoted  by  Kw,  is  the  product  of  the  equilibrium 
concentration  of  H'  and  OH'  in  pure  water.     That  is  — 

KW  =  CeH'    X    CeOH' 
CcHoO 

so  that  K!  =  ~ 


MEASUREMENT  OF  AFFINITY  347 

In  the  case  under  discussion  the  H'  ion  in  the  acid  and 
the  OH'  in  the  alkali  are  both  normal  (i.e.  unity),  dissociation 
being  assumed  complete  for  the  sake  of  simplicity,  so  that  their 
logarithms  in  the  second  term  vanish.  Hence  we  can  write  — 

A  =  RT  log,  p-^  ----  RT  log,  CH,o 

C€H-  X  CfcOH' 

Further,  since  in  any  dilute  aqueous  solution  we  can  re- 
gard the  concentration  of  the  water  to  be  practically  constant, 
the  term  CH/)  =  CeH2o,  so  that  — 

A  =    -RT  log*  C.H-  X  C6oH'  =    -RT  log*  Kw 
.-.  o'8i  =  —0-86  X  io~4  X  290  X  2-303  Iog10  Kw 
whence   K«,  =  io~14 

This  agrees  very  well  with  the  ionisation  constant  of  water 
obtained  by  other  means  (conductivity,  hydrolysis,  catalysis). 
By  taking  into  account  the  solution  pressure  of  hydrogen  at 
the  hydrogen  electrode,  and  treating  the  system  simply  as 
a  concentration  cell  in  respect  of  H*  ions,  we  are  led  to  the 
same  result.  Thus  denoting  by  Pn2  the  solution  pressure  of 
hydrogen,  we  can  write  the  e.m.f.  E  as  — 

RT        _  PH,_  RT     g/H-  in  N  acid 

~  »F     §/H-  in  normal  alkali  "^  «F     g          Pn2 
_  55         £H;  jn_ac_id  _  RT         CH-  in  acid 
"  »F    °g/H-ln  alkali  -   «F    °S  CH-  in  alkali 
But  CH-  in  the  acid  is  unity,  hence  — 

RT 

E  —  ---  ^  log  CH-  in  alkali 
nr 

Now  under  all  circumstances  (i.e.  in  any  dilute  aqueous 
solution)  it  is  considered  that  Cir  X  COH'  =  Kw  (the  ionisation 
constant  of  water),  so  that  — 

E  =  -Hlog   K»  in  alkali 
nb 


and  since  COH'  in  the  normal  alkali  is  also  unity  — 

RT. 
-^loglw 

But  A  =  nEV      .'.  A  =  —  RT  log  KIt,  as  before. 


348        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

The  affinity  of  H'  at  normal  concentration  for  OH'  also 
at  normal  concentration,  in  aqueous  solution,  amounts  to 
—  RT  log  Kw  or  +  0-8 1  volt-faradays,  or  18,620  calories  per 
gram-ion  or  per  gram-mole  of  water  formed.  The  heat  of  the 
same  reaction,  which,  it  has  already  been  shown,  is  necessarily 
the  heat  of  neutralisation  of  one  equivalent  of  a  strong  acid 
with  a  strong  alkali,  is  14,000  calories  approximately. 

The  same  result  would  have  been  obtained  if  we  had  used 
an  oxygen  electrode  in  both  solutions,  only  that  here  the 
direction  of  the  current  would  have  been  reversed.  A  third 
method  consists  in  combining  an  oxygen  and  hydrogen 
electrode,  thus — 

H2    |    Normal  HC1    |    O2 

Any  other  electrolyte  may  be  used,  but  is  not  so  good  in  prac- 
tice as  acid.  In  this  cell  hydrogen  dissolves  in  the  ionic  state 
as  also  does  the  oxygen  as  OH',  thereby  giving  water.  Both 
gases  are  therefore  used  up  in  the  process.  The  results  can  be 
calculated  as  before  (see  Sackur,  /.<:.,  pp.  92,  93).  Perhaps  the 
simplest  method  is  to  assume  the  correctness  of  the  standard 
calomel  electrode  and  measure  the  e.m.f.  of  the  combination — 

N°arOH         Calornel 

or  better  still  by  using  Allmand's  alkaline  standard  half 
element  Hg  |  HgO,  thus  obtaining  the  single  potential 
H2  |  NaOH,  from  which,  knowing  the  solution  pressure  of 
hydrogen  or  its  "  electrolytic  potential,"  the  concentration  of 
H*  ions  in  the  alkali  can  be  calculated  and  hence  the  dis- 
sociation constant  of  water. 

The  hydrogen-oxygen  or  Knall-gas  cell,  as  it  is  often  called, 
offers  a  very  striking  illustration  of  the  inapplicability  of  the 
Berthelot  principle  of  considering  heat  as  the  measure  of 
affinity. 

Thus  in  the  case  of  the  cell, 


H2    |     Normal  NaOH     |     O2 

as  already  pointed  out,  hydrogen  dissolves  in  the  ionic  state  in 
the  alkali,  thereby  leaving  the  pole  negative.     Oxygen  also 


MEASUREMENT  OF  AFFINITY  349 

dissolves,  leaving  the  oxygen  pole  positive.     If  E  is  the  total 
electromotive  force  of  the  cell  — 


tog      .  +         lo         , 

n       to/H-        «i        Am' 

p«       pttj 

E  =  RTlog^-^ 
/H-  *  A>H' 

(Neglecting  the  liquid  P.D.)  n  =  2  for  H2  =  2H'  and  n±  =  4 
for  O2  =  2O"  =  4OH'. 

Evidently,  therefore,  if  the  gases  oxygen  and  hydrogen, 
instead  of  being  fed  into  the  cell  at  atmospheric  pressure,  are 
introduced  at  a  lower  pressure,  the  e.m.f.  of  the  icell  will 
also  be  lower.  [The  solution  tensions  PO.,  and  Pn2  will  vary 
with  the  pressure  at  which  the  gases  are  maintained,  since  their 
solubility  in  the  platinum  varies  with  pressure.]  In  fact  if  the 
pressures  of  the  gases  be  reduced  almost  to  zero  the  e.m.f. 
will  almost  disappear.  Under  such  circumstances  water  may 
evidently  be  decomposed  by  currents  at  minimum  e.m.f.,  it 
being  only  necessary  to  apply  one  which  exceeds  that  of  the 
cell  itself  by  a  very  small  amount.  It  is  clear  from  this  that 
the  electrical  energy  obtainable  through  the  formation  of  water 
from  oxygen  and  hydrogen  or  necessary  for  its  decomposition 
(the  two  being  equal  and  of  opposite  sign)  may  assume  any 
magnitude  from  zero  to  a  certain  value  dependent  on  the 
pressures  of  the  gases  or  their  concentrations  in  the  platinum. 
The  heats  of  formation  of  water  at  constant  pressure,  on  the 
other  hand,  are  the  same  no  matter  at  what  pressure  we  work, 
and  this  is  the  most  direct  evidence  that  a  simple  relation 
cannot  exist  between  the  heat  of  reaction  and  the  electrical 
energy  obtained.  (Le  Blanc,  Electrochemistry,  p.  255.  Le 
Blanc  further  shows  that  this  decomposition  at  minimum 
e.m.f.  is  not  in  contradiction  to  the  Second  Law.) 

Affinity  of  Oxidation  and  Reduction  Processes. 

The  electrical  method  of  measuring  the  affinity  of  oxida- 
tions and  reductions  which  can  be  set  up  in  the  form  of  a  cell, 
is  identical  with  former  affinity  measurements,  consisting  again 


350        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 


in  the  determination  of  the  e.m.f.  If  into  an  aqueous  solution 
containing,  say,  ferro  and  ferri  salts  an  electrode  be  introduced 
which  must  not  be  attacked  by  any  of  the  components  taking 
part  in  the  reaction  (say,  a  platinum  electrode)  but  is  simply 
employed  as  a  carrier  of  current  to  and  from  the  cell,  and  if 
further  a  hydrogen  or  a  calomel  electrode  be  employed  as  the 
other  half-element,  then  we  have  a  cell  such  as : 

Fe+++     : 

Pt  J®++  \  ti- 
the e.m.f.  of  which  is  a  measure  of  the  oxidation  or  reduction 
process  taking  place.  In  the  particular  case  mentioned,  in 
which  the  hydrogen  electrode  is  used  as  one  half  element,  the 
total  reaction  occurring  in  the  cell  may  be  represented  by 

H2  +  2Fe+++->2H+ 
i.e.  a  reduction  of  ferri  to  ferro, 
or  2H+  -j-  2Fe^+->H2  - 

t.c.  an  oxidation  of  ferro  to  ferri. 

The  reaction  in  the  two  parts  of  the  cell  may  also  be 
represented  thus 

.2®  H2  —  2®  ^Reduction  current 

=  2Fe++  formed         =  2H'  formed     )     right  to  left 

(-}-  2®  means  that  two  electrons  come  from  the  platinum  to 
the  ferri-ferro  solution,  and  —  2®  on  the  hydrogen  side  denotes 
that  two  electrons  are  given  up  to  the  hydrogen  electrode,  the 
two  effects  occurring  simultaneously.) 

or  2Fe+++  +  H2  =  2H'  -f  2Fe++ 

Similarly  the  oxidation  process,  with  current  flowing  from 
left  to  right,  can  be  represented — 

—  2®     :     2H'  4-  2® 


or  2Fe++  +  2tT  =  2Fe+++  +  H2 

The  equilibrium  constant  of  the  above  reduction  reaction 


s  — 


:    |>Fe"]«X 
[H2]  X  [F( 


MEASUREMENT  OF  AFFINITY  351 

square    brackets    denoting    concentration   terms.       And   the 
affinity  A  is — 

A  =  RT  log  K  -  RT  log  ^£^1^ 

A  is  the  affinity  per  mole  of  hydrogen  (H2).     The  e.m.f.  E  is 
the  affinity  per  gram  equivalent,  i.e.  E  = 

RT  RT        [Fe++]2  X  [H']2 

2     og  2     og  ,      ,       rFe+++-i2 


T?nr 

K^  log  K  =  E'o 


E0  is  the  value  of  E  of  the  cell  when  the  Fe++  and  Fe+++ 
are  at  the  same  concentration  in  the  solution.  E0  is  called 
the  normal  potential  of  the  process.  E  is  called  the  reduction 
potential.  Of  course,  if  we  had  been  considering  the  reverse 
process,  i.e.  an  oxidation,  we  would  have  written 


and  would  finally  have  obtained  — 


or 

Reversible  reduction  and  oxidation  processes  are  always 
thus  connected.  Suppose  that  the  actual  conditions  in  the 
cell  itself  are  such  that  a  reduction  is  going  on,  yielding  an 
e.m.f.  of  E.  Then  if  an  external  e.m.f.  —  E  be  applied  in 
the  opposite  direction,  the  reduction  process  will  stop,  and  on 
making  the  externally  applied  e.m.f.  just  greater  than  the 


352        A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

• 

direct  e.m.f.,  an  oxidation  will  take  place.  The  above  formula 
brings  out  the  dependence  of  the  oxidising  force  upon  the 
concentration  of  the  various  components  which  are  taking 
part  in  the  reaction. 

R.  Peters  (Zeitsch.physik.  Chem.^  26,  193,  1895)  was  tne  nrst 
to  systematically  examine  oxidation  and  reduction  processes  in 
such  cells  under  definite  conditions.  He  used  as  one  half  of 
the  element  the  normal  calomel  electrode,  the  value  of  which 
in  absolute  measure  was  taken  to  be  0-56  volts,  the  mercury 
being  positive  with  respect  to  the  solution.  The  cell  might 
thus  be  diagrammatically  represented — - 


Pt 


Fe 


Hg2Cl2 


Suppose  that  the  process  occurring  is  a  reduction  of  some 
of  the  ferri  to  ferro  ions.  The  direction  of  the  inside  current 
will  then  be  from  right  to  left  ;  or  what  is  the  same  thing,  the 
direction  of  electron  transfer  will  be  from  the  platinum  to  the 
mercury.  The  reaction  in  each  half  of  the  cell  is  thus  — 


20 


=  2Fe++  (formed) 


Hg2  —  2® 

Metal 

=  Hg2++  formed 


These    processes    can    be    represented   by   the    chemical 
equation — 

+  +  Hg2  =  Hg2+*  +  2Fe++ 


Metal 


The  equilibrium  constant  K'  is  thus — 


Metal 


or  since  CHg2++>  and  likewise  Cng2  metal,  are  constant,  we  can 
write — 


and  therefore — 

A  =  RT  log  K 


RT  log  -^ 


MEASUREMENT  OF  AFFINITY 


353 


Since  A  is  reckoned  per  mole  of  mercury  ionised,  the 

A 

e.m.f.  E  =  -,  or — 


T-«  1  ir  TTT«  1 

E  =  —  log  K  —  RT  log 


RT 


setting  —  log  K  =  E0,  we  obtain — 
E  =  E0  —  RT  log 


Peters  obtained  a  large  number  of  results  with  ferric  and 
ferro  mixtures,  as  well  as  with  other  ic  and  ous  ions.  The  follow- 
ing are  a  few  of  his  results.  The  mixtures  were  made  from  a 

N                                  N 
stock  solution  of  —  ferrous  chloride-] hydrochloric  acid 

(the  acid  being  added  to  prevent  hydrolysis),  and  a  similar 

N  N 

solution  of  —  ferric  chloride  -f-  —  hydrochloric  acid.     In  the 

experimental  results  quoted  the  current  flows  inside  the  cell 
from  calomel  to  the  platinum,  that  is,  electrons  pass  from  the 
platinum  to  the  ferro-ferri  solution,  and  therefore  reduction 
takes  place. 


Ratio  of  Fe++,  and 
Fe+++  in  %• 

E=obs.  e.m.f.  of 
cell  (platinum  the 
positive  pole)  in 
volts. 

P.  D  of  the 
ferri  |  ferro 
electrode 
in  volts. 

E0  calculated 
from  E  obs.  by 
applying  the 
work  equation. 

K  calculated 

_  ceFe++ 

C€pe++  + 

Ferri.      Ferro. 

0'5       99-5 

0-296 

0-856 

0-428 

0-312 

0-872 

0-427 

2            98 

0-33I 

0-891 

0-428 

Mean  value 

10         90 

50            50 

Q'375 
0-427 

0-935 
0-987 

0-430 

0-427 

I07MS 

90         10 

0-483 

1-043 

0-428 

99           i 

0'534 

1-094 

0-419? 

The  values  of  E0  are  calculated  on  the  assumption  that 
the  concentration  ratio  of  the  ions  Fe++  and  Fe+++  is  the  same 
as  the  ratio  of  the  ferri  and  ferro  salt  concentration,  which 
of  course  can  only  be  regarded  as  an  approximation.  The 

T.P.C.— II.  2  A 


354       A   SYSTEM^  OF  PHYSICAL   CHEMISTRY 

applicability  of  the  theory  is,  however,  demonstrated  by  the 
constancy  of  the  value  E0.  The  numerical  value  of  K  will  be 
seen  to  be  very  great.  This  means  that  when  the  equilibrium 
is  reached  in  ferro-ferri  ion  solution  the  ratio  of  ferro  to 
ferri  must  be  very  great,  viz.  over  10  million.  It  will  thus  be 
evident  that  in  Peters'  mixtures  (even  in  that  containing  only 
o'5%  Fe+++)  there  is  too  great  a  concentration  of  Fe+++,  and 
there  is  the  tendency  for  reduction  to  take  place. 

VARIATION  OF  AFFINITY  WITH  TEMPERATURE. 

The  variation  of  A,  that  is  to  say  of  nE  when  the  reaction 
occurs  in  a  cell  (and  n  represents  the  number  of  faradays 
associated  with  the  unit  of  mass  considered),  is  easily  found 
by  simply  applying  the  Gibbs-Helmholtz  equation — 


since  -f-  U  =  decrease  in  internal  energy  —  heat  evolved  at 
constant  volume  =  —  Q«  on  the  notation  previously  employed. 
On  substituting  «E  for  A  we  obtain  — 


an  expression  giving  the  connection  between  the  e.m.f.  of  a 
cell  and  the  heat  of  the  chemical  reaction  occurring  in  the 
cell,  if  the  reaction  were  carried  out  in  a  calorimeter,  and  the 
heat  actually  measured.  This  expression  gives  us  the  true 
relation  of  heat  evolution  to  affinity.  In  a  particular  case, 
if  the  e.m.f.  has  a  negligibly  small  temperature  coefficient 

^E 

—  =  o,  approx.,  e.g.  the   Daniell   cell,  the   e.m.f.  becomes 

numerically  equal  to  the  heat,  i.e.  under  this  condition  affinity 
and  heat  evolution  are  identical  —  Thomson's  Rule  and  Berthe 
lot's  Law. 


MEASUREMENT  OF  AFFINITY 


355 


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356        A    SYSTE^  OF  PHYSICAL   CHEMISTRY 

The  last  two  columns  should  theoretically  be  identical, 
their  agreement  is  verification  of  the  Gibbs-Helmholtz  rela- 
tionship. The  last-mentioned  cell  is  particularly  interesting 
in  that  the  heat  of  the  reaction  is  negative,  although  the  same 
reaction  can  give  rise  to  a  positive  electrical  energy  output. 
In  this  case  the  Berthelot  Law  absolutely  breaks  down. 


CHAPTER   XII 

Systems  not  in  equilibrium  (continued) — Relation  between  the  affinity  and 
the  heat  of  a  reaction — Nernst's  heat  theorem  and  some  of  its 
applications . 

THE  RELATION  OF  TOTAL  ENERGY  TO  FREE  ENERGY. 

The  Thermodynamical  Theorem  of  Nernst.  (Cf.  Nernst's 
Theoretical  Chemistry,  English  Translation  of  the  6th 
German  Edition,  p.  709.  Also  more  particularly  Nernst's 
"  Applications  of  Thermodynamics  to  Chemistry,"  Silli- 
man  Lectures,  1906;  also  for  a  compendium  of  more 
recent  results  see  the  article  by  F.  Pollitzer,  Ahrens 
Sammlung,  vol.  17,  1912.) 

The  fundamental  problem  of  greatest  interest  in  the 
domain  of  chemical  energetics  is  that  of  the  quantitative 
relation  of  the  total  or  internal  energy  (decrease)  U  involved 
in  a  reaction  to  the  free  energy  A,  involved  in  the  same 
reaction.  Since  the  total  energy  change  is  identical  with  ±  Qr, 
the  heat  absorbed  or  evolved  in  the  reaction  at  constant 
volume,1  the  problem  may  be  put  in  the  slightly  different  way  : 
What  is  the  quantitative  connection  between  the  free  energy 
change  and  the  heat  change  in  any  given  reaction,  that  is, 
is  it  possible  to  calculate the  free  energy  change  from  a  measurement 
of  the  heat  change  ?  We  have  already  briefly  considered  this 

1  Reactions  in  general  do  involve  a  volume  change.  Suppose  in  the 
reaction  in  question  the  volume  increases ;  we  measure  this  volume 
increase  and  multiply  by  the  pressure  under  which  the  expansion  took 
place,  thereby  obtaining  a  work  term  which  can  be  expressed  in  calories, 
say.  This  work  W  must  have  been  done  at  the  expense  of  the  internal 
energy  of  the  system,  and  hence  if  the  observed  heat  (per  gram  equivalent) 
is  Q,  the  value  which  it  would  have  had  if  no  volume  change  had  taken 
place,  namely,  Qv,  is  given  by  the  equation  Q«  =  Q  4-  W. 


358        A    SYSTEM'OF  PHYSICAL   CHEMISTRY 

question  in  the  foregoing  chapter.  The  earliest  and  simplest 
view  held  by  Joule  and  Thomson  was  that  the  two  terms  were 
identical,  namely,  that  A  =  —  Qv.  They  found  this  conclusion 
fairly  accurately  borne  out  in  the  case  of  the  reaction 
Zn  +  CuSO4  ->  ZnSO4  -f  Cu  which  takes  place  in  the  Daniell 
cell,  but  this  agreement  must  be  looked  upon  as  accidental, 
due  to  the  fact  that  the  electromotive  force  of  this  cell 
happens  to  have  practically  no  temperature  coefficient.  The 
simple  equality  breaks  down  completely  in  many  other  cases, 
cf.  the  foregoing  table.  We  have  already  discussed  briefly 
this  Berthelot  principle  in  connection  with  affinity.  In 
many  cases,  especially  in  reactions  between  solids  and  liquids 
(i.e.  the  so-called  condensed  systems),  the  Berthelot  principle 
holds  fairly  well,  but  breaks  down  completely  in  reactions 
occurring  in  gases  or  dilute  solutions.  To  show  the  approxi- 
mation to  quality  between  the  two  terms  A  and  Qr,  one  may 
cite  the  following  instances  of  reactions  which  take  place  in 
condensed  systems  capable  of  being  set  up  in  the  form  of  cells 
and  therefore  permitting  the  direct  measurement  of  A  inde- 
pendently of  the  direct  measurement  of  Q?,. 


Reaction. 


A  volt- 
faradays.1 


2Hg  +  PbCl2  ->  Pb  -f  2Hg2CU  i     0-54 
2Ag  +  PbCl2  ->  Pb  +  2AgCl  "        0-49 
" 


2Ag-fL,       "~>2AgI  0-6 

Pb  +  L,  "         -»  PbI2  0-89 


Qv  volt- 
faradays. 


0-44 
0-52 
o'6o 
0-87 


These  values  hold  good  for  ordinary  temperatures,  but  even 
in  the  case  of  condensed  systems  the  approximate  equality 
vanishes  if  we  work  at  higher  temperatures.  Perhaps  the  most 
striking  exception  to  the  principle  of  the  equality  of  A  and 
Qw  or  U  is  that  furnished  by  the  process  of  fusion  or  transition 

1  A  being  an  energy  or  work  term  should  be  represented  in  electrical 
terms  by  the  product  of  volts  x  quantity  of  current.  In  the  above  cases 
I  faraday  is  supposed  to  have  passed  through  the  cell  corresponding  to 
I  gram  equivalent  of  the  reacting  metals.  This  is  assumed  in  the 
numerical  values  quoted.  Qt,  is*  first  obtained  in  calories  and  then  con- 
verted into  volt-faradays. 


AFFINITY  AND  HEAT  OF  REACTION        359 

from  one  phase  to  another.  At  the  melting  point  A  is  prac- 
tically zero  (the  specific  volume  of  the  solid  and  liquid  being 
generally  pretty  close  to  one  another),  whilst  Qv  is  usually  a 
large  number.  The  only  accurate  relation  between  A  and  Qv 
or  U  is  that  furnished  by  the  Gibbs-Helmholtz  equation  based 
on  the  First  and  Second  Laws  of  Thermodynamics.  This,  as 
we  have  just  seen,  is — 


When  the  free  energy  of  the  reaction  is  independent  of 
temperature  (as  is  nearly  the  case  in  the  Daniell  cell)  the 
terms  A  and  U  become  equal  to  one  another.  They  must 
likewise  be  equal  to  one  another  at  the  absolute  zero  of 
temperature  (T  =  o)  even  when  the  free  energy  does 
possess  a  temperature  coefficient.  These  conditions  show  the 
restrictions  necessary  to  be  applied  to  the  principle  of 
Berthelot.  The  principle  is  absolutely  true  for  all  reactions 
at  the  absolute  zero.  This,  however,  as  one  can  see,  is  of 
little  practical  value  or  importance.  The  fact  that  the  prin- 
ciple holds  even  approximately  at  ordinary  temperatures  is 
due  to  the  fact  that  our  ordinary  temperature  is  not  so  far 
removed  from  the  zero.  Although  this  principle  as  an  accurate 
physical  law  can  no  longer  be  relied  upon,  nevertheless  it 
would  be  in  the  highest  sense  unscientific  to  discard  it  alto- 
gether. It  is  to  Nernst  that  we  owe  the  most  successful 
solution  of  the  problem  of  finding  out  what  is  the  correct  law 
underlying  the  approximately  true  principle  of  Berthelot.  To 
appreciate  Nernst's  thermodynamic  theorem,  as  it  is  called,  let 
us  return  for  a  moment  to  the  Gibbs-Helmholtz  equation — 


In  cases  such  as  reactions  occurring  in  cells  in  which  it  is 
easy  to  measure  A  and  -==,  directly,1  this  equation  is  sufficient 
to    allow   us   to  calculate  U,   but   on   the   other    hand   the 
1  In  this  differential,  volume  is  supposed  to  be  kept  constant. 


360       A   SYSTEM  JO F  PHYSICAL    CHEMISTRY 

equation  is  not  sufficient  to  allow  us  to  carry  out  the  reverse 
process,  namely,  that  of  calculating  A  from  an  observation  of 
U.  It  is,  therefore,  not  a  complete  answer  to  the  problem 
which  we  have  set  before  us.  The  limitations  to  the  informa- 
tion obtainable  from  this  equation  may  be  shown  in  a  some- 
what different  way  if  we  integrate  the  expression.  After  a 
slight  transformation  the  equation  can  be  written  in  the  form — 

T  — _A=-U 

Vr 


or 

which  is  the  same  as  writing — 


T2 

On  integrating  this  between  the  temperature  limits  o  and  T, 
we  obtain — 


where  a  is  an  undetermined  integration  constant.  Now  it  has 
been  shown  by  experience  that  the  heat  evolved  or  absorbed, 
that  is  ±  U,  in  any  reaction  which  takes  place  in  a  condensed 
system  (solid  or  liquid  (note  the  restriction))  varies  with  the 
temperature  at  which  the  reaction  is  made  to  take  place,  and 
this  variation  can  be  expressed  by  a  series  of  terms  involving 
T  raised  to  different  powers,  that  is  for  reactions  between 
solids  or  liquids  we  can  write  — 


U  =  U0  +  aT  +  j3T2  +  yT3  +  8T*  etc. 

U0  being  the  value  of  U  at  the  absolute  zero.     Substituting 
this  expression  for  U  in  the  integral  one  obtains  — 

|=  a  4-  ^  _-  a  log  T  -  )8T  ~  ^T2  _  ifiTS-,  etc. 
or  A  =  U0-KT—  aTlogT—  jST2—  jT3-  -T*—  ,  etc.    (i) 

9 


NERNST^   HEAT   THEOREM  361 


This  expression  gives  A  in  terms  of  U  or  Qv,  but  there 
still  remains  the  undetermined  integration  constant  a  to  be 
reckoned  with.  It  may  be  noted  that  the  numerical  values 
for  the  coefficients  a,  )3,  y,  etc.,  are  most  easily  obtained  from 
measurements  lof  the  specific  heats  of  the  solid  or  liquid 
reactants  and  resultants  in  any  given  reaction  which  occurs  in 
a  condensed  system  by  making  use  of  the  Kirchhoff  Law, 
namely  — 

^     r      c 
^=ci-c* 

where  Cj  is  the  heat  capacity  of  the  system  before  transforma- 
tion, C2  that  after  the  transformation. 

Since    U  =  U0  +  aT  +  j3T2  +  yT3  -f-  .  .  . 


The  integrated  form  of  the  Gibbs-Helmholtz  equation,  even 
when  restricted  to  reactions  in  condensed  systems  (which 
restriction  allows  of  the  expression  of  U  in  terms  which  can 
be  experimentally  determined)  is  still  not  sufficient  to  solve 
completely  the  question  of  the  quantitative  relation  between 
A  and  U  ;  there  remains  the  integration  constant  a.  The  real 
significance  of  Nernsfs  Theorem  lies  in  this,  that  it  permits  us  to 
evaluate  the  integration  constant  a. 

The  theorem  of  Nernst  may  be  stated  in  the  following 
way  :  Not  only  are  A  and  U  identical  at  the  absolute  zero  of 
temperature  itself  (as  the  Gibbs-Helmholtz  equation  requires), 
but  this  equality  also  holds  true  for  a  short  region  above  and  in 

the  neighbourhood  of  zero,  the  curves  for  -^  and  -=z  coinciding 

for  a  short  region  and  not  simply  touching  one  another  at  the 
absolute  zero  point  only.  This  means  that  for  a  short  range 
in  the  neighbourhood  of  zero,  A  and  U  remain  constant  and 
identical.  This  theorem  of  Nernst  may  be  expressed  graphi- 
cally as  in  the  figure  [Fig.  89],  in  which  the  change  of  A  and 
U  with  T  is  represented.1  In  the  upper  part  of  the  diagram  is 

1  In  the  figure  it  is  to  be  supposed  that  the  U  and  A  lines  run  together 
and  are  horizontal  for  some  distance  above  the  zero  value  of  T. 


362        A   SYSTEM  GF  PHYSICAL   CHEMISTRY 


represented  the  case  in  which  A  falls  with  rising  temperature 
while  U  rises,  in  the  lower  part  an  instance  of  the  reverse 
behaviour  (which  is  also  found  in 
u/  practice)  is  represented.  Stated,  ana- 

lytically, Nernst's  Theorem  is — 


T-    - 

Limit  -    =  o 


. 

and  Limit  -r^  =  o 
a  1 


for  T  =  o 


FIG.  89. 


This  easily  grasped  and  certainly 
not  unreasonable  supposition  has  been 
shown  by  Nernst  to  be  the  necessary 
and  sufficient  condition  for  enabling 
us  to  calculate  the  integration  con- 
stant a. 

Thus,  if  we  differentiate  equa- 
tion (T)  with  respect  to  T  we 
obtain — 


and  differentiating  the  expression — 

U  =  U0  +  aT  +  j8T2  +  yT3  +  ST* 
one  obtains — 

JTT 

.      .      (2A) 


Now  introducing  the  Nernst  Theorem— 


T  =  ^f  =  oor 
and  in  the  neighbourhood  of  zero,  it  follows  that  — 

a  -  a  -  a  log  T  -  2j8T  -  |yT2  -  f  8T3  =  0 
and  a  +  2jST  +  3j8T2  +  48T3  =  o 

In  order  that  the  two  expressions  may  simultaneously  be 
equal  to  zero,  it  is  necessary  that— 

the  integration  constant  a  =  o 
and  also  that  a  =  o 


NERNSTS  HEAT  THEOREM        363 

Hence,  allowing  for  this  in  equations  (2)  and  (3),  one 
obtains  finally — 

,ete..     .     (3) 
....     (4) 

Equations  (3)  and  (4)  are  the  solution  to  the  problem 
regarding  the  calculation  of  A  from  determinations  of  U> 
that  is  from  heat  measurements  alone,  i.e.  Qv  and  specific 
heats. 

The  coefficients  j8  and  y  may  be  either  positive  or  negative, 
and  hence  we  have  the  two  possibilities  respecting  the  direction 
of  the  >  slope  (up  or  down)  of  the  curves  for  A  and  U  shown 
in  the  diagram,  where  A  and  U  are  written  as  functions  of 
temperature  T. 

As  a  result  of  calculation  of  A  by  the  help  of  the  above 
equations  (3)  and  (4)  from  observed  values  of  U,  it  has  been 
found  that  in  a  great  many  cases  the  coefficients  j8,  y,  and  8 
are  small.  Hence,  if  the  heat  evolution  in  a  given  reaction  is 
great,  i.e.  if  U  is  great,  both  A  and  U  tend  to  have  almost  the 
same  value  (U0  approx.).  It  is  this  fact  which  lies  at  the  basis 
of  Berthelot's  principle. 

For  the  present  we  have  of  course  to  restrict  the  applica- 
tions of  equations  (3)  and  (4)  to  the  case  of  solid  or  liquid 
systems,  for  at  the  absolute  zero,  or  in  its  neighbourhood, 
gases  have  no  possible  existence.  This  restriction  is,  how- 
ever, not  of  such  importance  as  it  seems,  for  it  is  possible  by 
the  aid  of  the  first  two  laws  of  thermodynamics  to  calculate 
the  affinity  of  a  reaction  occurring  in  a  gaseous  system  (or  in 
a  dilute  solution)  if  we  know  the  affinity  and  heat  relations  for 
the  same  reaction  when  it  occurs  in  the  solid  state. 

It  must  be  remembered  that  the  Nernst  hypothesis  is  one 
which  we  cannot  verify  by  direct  experiment,  for  the  simple 
reason  that  the  measurement  of  A  and  U  in  the  neighbour- 
hood of  the  absolute  zero  cannot  be  carried  out.  We  can, 
however,  reach  very  low  temperatures  experimentally,  and 
Nernst  considers  an  extrapolation  of  such  as  justifiable. 

Perhaps  the  most  direct  evidence  which  shows  the  rapid 


364       A    SYSTEM  dF  PHYSICAL   CHEMISTRY 

fall  of  specific  heat  with  fall  in  temperature  is  offered  by  the 
two  systems,  benzophenone  and  betol.  Both  these  substances 
show  in  a  marked  degree  the  phenomenon  of  super-cooling, 
so  that  it  is  possible  to  measure  the  specific  heats  of  the 
solid  and  of  the  liquid  forms  at  temperatures  considerably 
below  the  true  melting  point.  The  total  energy  change 
(decrease)  in  passing  from,  say,  the  liquid  to  the  solid  state 
at  any  temperature,  is  represented  by  the  symbol  U,  such 
a  process  being  entirely  analogous  to  any  chemical  reaction 
in  which  a  substance  A  passes  into  another  B.  We  have  seen 
from  the  Kirchhoff  Law  that — 


According  to  Nernst's  Theorem  ^,   and    therefore   the 

difference  of  the  specific  heats  of  the  liquid  and  solid,  tend  to 
decrease  towards  zero  as  the  temperature  falls.  This  is  borne 
out  by  the  following  experimental  values  of  Koref. 


BENZOPHENONE. 

T. 

Specific  heat  of 
liquid. 

Specific  heat  of 
solid. 

f  =  Cl-c, 

295 
137 

0-3825 
0-1526 

0-3051 
0-1514 

0-0774 
O"OOI2 

BETOL  (INTERPOLATED  GRAPHICALLY). 

320 
240 
130 

0-362 
0-256 
0-148 

0-295 
0-2205 
0-144 

I 

0-067 
0-0355 
0-004 

To  illustrate  the  importance  of  Nernst's  Theorem  we  shall 
consider  a  few  well  known  reactions,  for  which  the  heat  effect 
U  is  known.  From  these  data  the  values  of  A  at  various 
temperatures  will  be  calculated  and  then  compared  with 
observed  values  of  A,  when  such  have  been  determined. 


NERNSTS   HEAT   THEOREM 


365 


APPLICATIONS  OF  THE  NERNST  THEOREM  TO  CONDENSED 
SYSTEMS. 

i. — The  Transformation  of  Rhombic  Sulphur  into  Monocline. 

Employing  equation  (3)  in  an  abbreviated  form,  one  finds 
for  the  total  energy  change  involved  in  the  transformation, 
say,  of  i  gram  of  rhombic  into  monoclinic  sulphur,  the 
expression — 

and  therefore — 

Cx  and  C2  being  the  specific  heats  of  the  two  modifications. 
From  measurements  of  these  specific  heats  it  is  found  that 
j8=ri5  x  io~5.  From  Bronsted's1  data  for  the  heat  of 
transformation  at  constant  volume,  i.e.  Qv  or  U,  one  obtains 
with  a  knowledge  of  )3  the  value  of  U0.  The  above  equation 
for  U  takes,  therefore,  the  form — 

U  =  1-57  +  1-15  X  io~&T2 

Nernst  (/.  <:.),  also  Nernst's  Text  Book  (English  translation 
of  6th  German  edition,  p.  713,  seq.)>  gives  two  instances  of  the 
application  of  this  formula. 


T. 

U  calculated. 

U  observed. 

Observer. 

Ill 

2-40 
3^9 

2'43 
3*i3 

Bronsted. 
Tammann. 

The  agreement  is  good. 

Further,  by  applying  Nernst's  second  equation  (equation  4), 
also  in  an  abbreviated  form,  we  have — 

A  =1-57  —  ri5  X  IO-&T2 

At  the  transition  temperature  the  free  energy  involved  in 

a  change  from  one  modification  to  the  other  is  zero,  since  the 

two  forms  are  in  equilibrium  (except  of  course  for  the  negligible 

amount  of  work  done  against  the   atmosphere,  due   to   the 

1  Zeitsch.  physik.  Chem.,  56,  645,  1906, 


366       A   SYSTEM  9F  PHYSICAL   CHEMISTRY 


slight  volume  change  involved).  By  putting  A  =  o  we  should 
be  able  to  calculate  the  transition  point  by  the  aid  of  the  above 
equation.  Denoting  this  temperature  by  the  symbol  T0  we 
have — 

Tn  = 


—  =369-5  abs. 
j*i5  X  io~5 

T0  observed  =  368*4  abs. 

Nernst  has  further  calculated  by  the  aid  of  the  expression 
for  A  the  values  corresponding  to  several  temperatures  for 
which  A  had  been  determined  by  Bronsted  by  means  of 
solubility  measurements.  The  following  table  contains  the 
values  thus  obtained. 

FREE  ENERGY  CHANGE  IN  CALORIES  INVOLVED  IN  THE  TRANSFOR- 
MATION OF  SULPHUR  FROM  MONOCLINIC  TO  RHOMBIC  FORM. 


T  absolute. 

A  calculated 
(Nernst). 

A  observed 
(Bronsted). 

273 

0-72 

071 

288-5 

0-64 

0-61 

291-6 

0*63 

o'59 

298-3 

0'57 

o'SS 

Nernst  has  further  shown  that  the  observed  values  of 


d\3 
df* 


viz.  (Ci  —  C2)  over  a  wide  temperature  range  down  to  83° 
absolute  is  fairly  accurately  produced  by  the  expression  2j3T. 
This  is  shown  in  the  following  table. 


T  absolute. 

^  observed. 

*T=,30X,0-, 

Observer. 

83 

0-0854 

-  0*0843  = 

0*001  1 

0*0019 

Nernst 

93 

0-0925 

-  0*09I5  = 

O'OOIO                 O'OO2I 

}> 

138 

0*1185 

-0-II3I  = 

0-0054                 0*0032 

Koref 

198 

0-1529 

—  0*1473  = 

O'OO56                :                               O'OO46 

Nernst 

235 

0-1621 

-0-1537  = 

0*0084            °'oo54 

Koref 

290 

0*1774 

—  0*1720  = 

0*0054            0*0067 

Wigand  1 

293 

0-1794 

-  0-I705  = 

0*0089            0-0067 

Koref 

299 
329 

0-1809 
0*1844 

-  0*I727  = 
-  0-1764  = 

0*0082            0*0069 
0*0080    i        0*0076 

Wigand 
Regnault 

Wigand  :  Annalen  tier  Physik,  [4]  22,  79. 


NERNSTS   HEAT   THEOREM  367 

The  comparison  of  theoretical  conclusion  with  experi- 
mental results  in  the  case  of  the  transformation  of  sulphur 
has  substantiated  Nernst's  Theorem  regarding  the  possibility 
of  calculating  A  from  purely  thermal  data  in  a  very  satisfactory 
way.  It  will  be  observed  that  as  T  increases  A  decreases, 
finally  becoming  zero  at  the  transition  point.  U,  on  the  other 
hand,  increases  as  T  increases.  The  behaviour  of  this  system 
is  thus  represented  by  the  curves  U  and  A  belonging  to  type  I. 
in  the  figure  (Fig.  89). 


2.  —  Application  of  Nernsfs  Theorem  to  the  Calculation  of  the 
Temperature  of  Fusion  of  Single  Substances. 

At  the  melting  point  the  solid  and  liquid  are  in  equilibrium, 
that  is  A  =  o  and  therefore  equation  (4)  becomes  — 

U_  8T2—  ^T3  =  o 


From  measurements  of  the  specific  heat  of  the  solid  and 
(super-cooled)  liquid  forms,  GI  and  C2  at  various  temperatures 
are  obtained  ;  and  from  these  one  obtains  values  of  j3  and  y 
from  the  expression  Cx  —  C2  =  2j8T  +  3yT2,  this  final  term 
being  generally  negligible.  From  these  data  U0  can  be 
calculated.  Employing  the  equation  — 


and  substituting  the  value  of  the  latent  heat  of  fusion  Qw  or 
U  at  the  melting  point  T,  the  temperature  (T)  required  can 
be  calculated. 


The  Process  of  Fusion. — The  free  Energy  and  total  Energy 
Changes  Involved. 

Some  examples  of  the  application  of  the  Nernst  Theorem 
in  a  slightly  different  form  to  that  already  followed  have 
been  investigated  by  J.  T.  Barker  (Zeitsch.  physik.  Chew.,  71, 
235,  1910)  in  Nernst's  laboratory.  The  following  table 


368        A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

contains  the  values  of  A  and  Qu  (the  latent  heat  of  fusion)  in 
the  case  of  benzene. 

FUSION  OF  BENZENE. 


Absolute  temperature. 

Latent  heat  of  fusion  in 
calories  per  mole. 

Free  energy  change. 

0 

940 

940 

5 

940-31 

939-69 

10 

941-23 

938-77 

20 

944'9 

93S'i 

50 

9707S 

909-25 

100 

1063 

817 

150 

1217 

663 

200 

H30 

45° 

250 

1710 

170 

g.     (ordinary  melt- 
7    4  \     ing  point. 

rnon  /observed  Q» 
1900  \2326  calories 

o 

The  relation  of  A  to  U  is  that  of  type  I.  (Fig.  89).  Other 
substances,  such  as  naphthalene,  show  a  closer  agreement 
between  Q  calculated  and  observed  than  does  benzene.  The 
paper  referred  to  should  be  consulted. 


3. — The  Affinity  of  Water  for  certain  Salts  in  the  production 
of  Hydrated  Salts. 

In  the  section  on  the  measurement  of  affinity  we  have  seen 
that  the  affinity  per  mole  of  a  salt  for  water  is  given  by  the 
expression — 

A  =  RT  log  ^ 

where  TT  denotes  the  vapour  pressure  of  water  at  the  tempera- 
ture T,  and  p  the  equilibrium  pressure,  i.e.  the  true  vapour 
pressure  of  the  salt,  that  is,  the  dissociation  pressure  of  the 
next  higher  hydrate.  To  obtain  A  by  the  aid  of  Nernst's 
theorem  from  purely  thermal  data  one  might  proceed  as 
follows.  First  of  all  obtain  the  heat  of  hydration  of  the  given 
salt.  This  has  been  done  very  comprehensively  and  accu- 
rately by  J.  Thomsen  (Thermo-chemistry  >  translated  by  A.  K, 


NERNST'S  HEAT  THEOREM        369 

Burke.  Sir  W.  Ramsay's  Series),  who  measured  directly  by 
calorimetric  means  the  heat  of  solution  of  a  given  hydrate, 
and  then  the  heat  of  solution  of  the  anhydrous  salt,  or  the 
next  lower  hydrate  formed.  The  difference  between  these 
two  heat  values  gives  the  desired  heat  of  hydration  in  the 
production  of  the  given  hydrate.  If  we  neglect  or  allow  for 
any  work  done  (at  the  expense  of  the  heat  energy)  due  to  a 
volume  change,  we  can  easily  obtain  Qv  or  U  for  the  given 
instance.  Now,  if  we  have  also  data  relating  to  the  specific 
heat  of  water,  of  the  hydrated  salt  in  question,  and  of  the  next 

lower  hydrate  or  anhydrous  form  we  should  know  ^  at  several 

temperatures,  and  hence  we  could  calculate  the  coefficients  j3 
and  y.  Knowing  U  itself  at  one  temperature  together  with 
j8  and  y  we  could  obtain  the  value  of  U0.  Now  knowing  U0, 
j3,  and  y,  equation  (4)  should  give  us  the  value  of  A  required 
at  once.  The  barrier  to  this  procedure  lies,  however,  in  the 
fact  that  the  specific  heat  of  water  is  abnormal  in  its  behaviour 
with  respect  to  temperature,  and  hence  Nernst  (Sitzingsber  d. 
Berlin  Akad.  d.  Wiss.,  52,  1906)  has  dealt  instead  with  the 
closely  allied  problem  of  the  affinity  of  ice  for  the  salt  in 
question,  since  the  specific  heat  of  ice  is  normal  in  behaviour. 
Its  molecular  heat  is  accurately  represented  by  the  equation  — 

(5) 


which  allows  us  to  extrapolate  to  temperatures  higher  than 
o°  C.,  i.e.  into  temperature  regions  where  we  would  naturally 
have  employed  the  specific  heat  of  water  had  this  been 
"normal."  The  values  of  Qw  obtained  by  Thomsen  refer 
to  the  heat  of  hydration  by  liquid  water.  We  must  convert 
this  into  heat  of  hydration  by  ice.  To  do  this  we  have 
to  substract  from  Thomsen's  value  the  value  of  the  molecular 
latent  heat  of  fusion  of  ice  at  the  given  temperature.  At 
o°C.  this  is  given  by  ^  =  1440  calories.  The  temperature 
coefficient  of  A  can  be  approximately  calculated  from  the 
difference  of  the  molecular  heats  of  water  and  ice  at  o°  C., 

namely,  -^  •    This  works  out  to  9*0  calories  per  degree.    Hence 
T.P.C.  —  ii.  2  B 


370        A   SYSTEM  dOF  PHYSICAL    CHEMISTRY 

at  1  8°  —  the  temperature  to  which  many  of  Thomson's  data 
refer  —  the  value  of  the  latent  heat  of  fusion  is  — 

A18o=  1440  +  (18  X  9*0)  =  1600  calories  per  mole 

Hence  the  value  of  U  required  —  the  total  energy  change  in 
the  process  of  adding  i  mole  of  ice  to  the  given  anhydrous  salt 
or  hydrate  —  is  easily  obtained  from  the  expression  — 

u  =  (QThomsen  ~  l6o°)  calories  per  mole 
Now  we  want  to  apply  the  theorem  to  calculate  A.     H. 
Schottky  (Zcitsch.  physik.   Chem.^    64,   415,   1908),  who   has 
investigated  several  cases  of  this  kind,  uses  equations  (3)  and 
(4)  in  their  simplest  forms,  viz.  — 

.....     (3A) 
.....     (4A) 


whence  =  d  —  C2  =  2j8T 

a  1 

The  reaction  in  general  may  be  written  — 

anhydrous  salt  +  H2OiCe  ->  hydrated  salt 
or  hydrated  salt  +  H2OiCe  ->  higher  hydrated  salt 

i.e.  (ABwH2O)  +  H2OiCe  ->  (AB(»  +  i)H2O) 

The  term  2J3T  is  the  difference  of  the  heat  capacities  of  the 
two  sides  of  the  above  equations.     Taking  the  second  case  — 

2/JT  =  (CABmHjjO  +  Cice)  —  CAB<W  +  1)H20 

Now  the  difference  of  the  specific  or  molecular  heat  terms 

CAB(m  +  l)H?OandCABmH20   1S   the    heat    CaPadty    °f    the    Water 

of  crystallisation   which   has   been   produced   from   the   ice. 
Denoting  this  by  C'  we  obtain— 

2j8T  =  Cice  -  C' 

If  we  are  dealing  with  the  case  of  hydration  of  the  anhydrous 
salt  by  the  addition  of  i  mole  of  ice  — 

2j3T  =  Canhydrous  salt  +  Cice  —  Cgait  HaO 

where  again  — 

CSalt  H.O  —  Cgalt  =  C'the  heat  capacity  of  the 
water  of  crystallisation 

so  that  2j3T  =  Cice  —  C' 


NERNSrs   HEAT   THEOREM  371 

Taking  as  a  particular  case  the  reaction — 

CuSO4  +  iH2O  ice  ->  CuSO4H2O 
Schottky  found  at  9°  C.,  from  equation  (5),  that — 

CiCe  =  9*29  calories  per  mole 
C'  =  6-99  (Schottky),  6«6o  (Fittig) 

and  hence,  using  Schottky's  C'  value — 

jq^j,  g 

2   X  (273  +  9) 

From  Thomsen's  data  U  at  18°  C.  is  4860  calories  per 
mole,  and  hence  from  equation  (3 A)  we  find — 

Uo  =  4520  calories  per  mole 
and  finally  from  equation  (40) — 

A  =  4520  — 0-00408X2 
and  therefore — 

AIS°C.  =  4180  |  calories  per  mole.     Using  Fittig's  value 
A78°  c.  =  4010  j       of  C7,  A78°  c.  =  3810  calories. 
Mean  Aygoc.  =  3910  calories  per  mole 

It  will  be  seen  that  A  at  18°  C.  is  less  than  U  at  the  same 
temperature.  They  were  identical,  of  course,  at  the  absolute 
zero,  the  relation  of  the  two  in  this  case  being  therefore  repre- 
sented by  curves  of  Type  I.  (Fig.  89). 

We  have  now  to  compare  the  value  of  A  obtained  above 
from  purely  thermal  data,  with  what  may  be  called  the  observed 
value  of  A,  namely,  that  obtained  from  vapour  pressure  measure- 
ments. The  dissociation  pressure/  of  CuSO4  *  H2O  has  been 
determined  at  78°  C.,  and  found  to  be  2-5  mm.  of  mercury.  (It 
is  owing  to  this  choice  of  temperature  that  it  was  necessary  to 
calculate  A  from  the  thermal  data  also  at  78°  C.)  Now  we 
have  to  obtain  the  vapour  pressure  TT  which  ice  would  possess, 
if  it  could  be  obtained  as  such,  at  78°  C.  This  is  obtained 
accurately  from  Scheel's  interpolation  formula1  (Verh.  d.  D. 
physik.  Gesell.,  8,  391,  1905),  viz. 

2687-4 


log  TT  =  11*4796  —  0*4  log  T  — 

J. 

1  The  experimental  verification  of  Scheel's  formula  must,  of  course, 


372       A   SYSTEM'OF  PHYSICAL   CHEMISTRY 


With  the  aid  of  this  formula  we  find 


Hence — 


=  639  mm.  Hg. 


A  observed  =  RT  log  -  =  -1-985  X  (273  +  78)  log 


or 


A  observed  =  3860  calories  per  mole 

Mean  value  of  A  from  thermal  data  =  3910  calories  per  mole. 

The  agreement  is  very  good. 

The  affinity  of  H2O  (in  the  form  of  ice  and  also  in  the 
liquid  form)  for  a  given  salt  varies,  as  one  might  expect,  accord- 
ing to  the  degree  of  hydration  already  possessed  by  the  salt. 
This  has  been  previously  pointed  out  in  the  section  on 
"  Affinity,"  where  the  relative  values  for  A  obtained  by  disso- 
ciation pressure  measurements  are  given  in  the  case  of  copper 
sulphate  and  its  hydrates.  The  molecular  heat  of  the  water 
molecules  also  differs,  from  molecule  to  molecule,  i.e.  it  is  not 
simply  an  additive  property.  The  following  table  illustrates 
this.  The  numerical  values  refer  to  the  H2O  molecule  which 
is  added  to  the  corresponding  salt  (producing,  of  course,  a 
higher  hydrate). 


Substance. 

Molecular  heat  (Schottky)  at 
9°C. 

Molecular  heat 
(Fittig). 

CuS04    . 
CuSO4lAq 
CuSO43Aq 
CuSO45Aq 
CuSOjAq 

24-09  calories 
3i-o8      „ 
48-88       „ 

67'I5       » 
88-3        „ 

25'07 
31-67 
47-92 
65-SS 
89-97 

refer  to  temperatures  below  o°  C.     The  following  three  observations  are 
sufficient  : — 


*°c. 

Mm.  Hg.    IT  calcu- 
lated by  Scheel. 

Mm.  Hg.    it  ob- 
served. 

0 
—  10 

-50 

4-58 

1-97 
0-031 

4-58 

1-97 
0-034 

NERNSFS   HEAT   THEOREM  373 

In  general  it  may  be  pointed  out  that  where  A  and  U  are 
very  nearly  the  same  —  as  is  the  case  with  some  of  the  salts 
above  mentioned  and  is  likewise  the  case  with  a  number  of 
reactions  which  can  be  set  up  in  the  form  of  voltaic  cells, 
e.g.  the  Daniell  cell  or  the  Clark  cell,  the  accurate  application 
of  the  Nernst  theorem  must  be  done  with  great  care  owing  to 
the  smallness  of  the  coefficients  j8  and  y  rendering  the  ex- 
pression (equations  (3)  and  (4))  particularly  sensitive  to  small 
errors  of  observation.  For  the  purpose  of  simply  testing  the 
theorem  it  is  only  fair  therefore  to  examine  those  cases  in 
which  A  and  U  differ  widely,  e.g.— 

CuSO4  +  H2O  ->  CuSO4  •  H2O. 

4.  —  Voltaic  Cells  consisting  of  Liquid  or  Solid  Substances. 
Calculation  of  the  Electromotive  Force  from  Thermal 
Measurements. 

In  the  present  case  we  are  confining  ourselves  to  cells  the 
constituents  of  which  are  either  solid  or  liquid,  since  the 
Nernst  Theorem  is  directly  applicable  to  these.  In  the  first 
instance  we  must  not  take  up  those  cases  in  which  solutions 
are  present,  since  at  absolute  zero  it  seems  probable  that  each 
phase  will  consist  of  a  single  substance  only.  Later  it  will  be 
pointed  out  how  the  theorem  may  be  extended  even  to  gas 
cells.  Convenient  types  of  cells  for  our  present  purpose  are 
the  following  — 

Cell.  Chemical  Reaction. 

Pb  |  PbCl2  |  AgCl  |  Ag  Pb  +  2  AgCl  ->  2Ag  +  PbCl2 

Solid.        Solid.          Solid.      Solid. 

Ag|  Agl|    I2  Ag  +  I2-»2AgI 

Solid.     Solid.     Solid. 

The  Clark  cell  at  the  cryohydric  point  — 
Zn|ZnS047H20|Hg2SO4|Hg     Zn  +  Hg2SO4 

Solid.  Solid.  Solid.       Liquid. 


Let  us  consider  the  case  of  the  Clark  cell.     This  cell  at 
ordinary  room  temperature  contains  the  zinc  salt  in  aqueous 


374       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

solution.  We  have  already  seen  that  it  is  desirable  to  work 
with  a  cell,  each  phase  present  being  a  single  "  pure "  sub- 
stance. To  realise  this  in  the  Clark  cell  we  must  work  at  the 
cryohydric  temperature,  namely,  — 9°C.,  at  which  ice  separates 
out.  The  reaction  is  indicated  above.  The  following  heat 
data  are  available  (at  17°  C.). 

Transformation  of  Zn  into  ZnSO4,  heat  evolved  230,090  cals. 
Transformation  of  ZnSO4  into  ZnS04'7H2O, 

heat  evolved 22,690     „ 

Total  heat  evolved  =     252,780    ',, 

Transformation  of  7  moles  of  ice  into  water 
of  crystallisation  =  approximately  to  heat 
of  fusion  per  mole  X  7  (heat  absorbed) 
=  (7X1580) =  1 1, 060  cals. 

Transformation  of  Hg2SO4  into  Hg2,  heat  ab- 
sorbed   =  175,000  „ 

Total  heat  absorbed  =     186,060     „ 

Therefore  the  nett  heat  evolved  per  mole  of  zinc  (say),  by 
the  above  reaction  =  66,720  calories.  This  value  holds  for 
290°  absolute.  This  is  the  total  energy  change  (neglecting  the 
very  small  volume  change)  per  mole  of  zinc  transformed.  If 
we  denote  by  U  the  total  energy  per  gram  equivalent,  then 
U  =  33,360  calories  at  290°  absolute. 

Now  we  have  to  calculate  j8  since  we  are  only  applying 
the  theorem  in  its  simplest  form,  viz.— 

U  =  U0  +  j3T2> 
A  =  U0  -  /3T2$ 

heat  capacity  of) (heat  capacity  of)  _  d\3  __    ~ 

reactants        j       (     resultants        5~~"^T~" 

The  following  data  on  the  molecular  heats  of  the  sub- 
stances concerned  are  quoted  by  Nernst  (Textbook,  English 
Translation  of  the  6th  German  Edition,  p.  747). 

Zn  =  6-0  (10°  C.),  Hg2S04  =  3ro  (50°),  7H2O  =  637  (10°), 
ZnS047H20  =  89-4(10°),  2Hg=i3-2. 


HEAT   THEOREM 


375 


From  these  figures  we  find,  on  reckoning  per  gram 
equivalent  instead  of  per  gram-mole  that  2j3T  =  —  0-95  for 
T  =  283  absolute.  These  values,  Nernst  points  out,  are  in 
several  instances  not  very  accurate,  and  further  they  do  not  all 
obtain  for  the  same  temperature.  They  must  suffice,  however. 
The  Nernst  Theorem  thus  gives  for  the  case  of  the  Clark  cell— 

U  =  38,505  —  o-ooiyT2 
A  =  38,505  +  ox>oi7T* 

Employing  this  formula  to  calculate  the  free  energy  per 
equivalent,  or  what  is  numerically  the  same  thing  the  electro- 
motive force  of  the  cell  in  volts,  one.  finds  for  the  cryohydric 
temperature  266°  absolute — 

E  =  1-4592  volts  (calculated) 

and  by  direct  observation  E  =  1-4624  volts  (observed).  The 
agreement  is  exceedingly  satisfactory.  Note  that  in  this 
reaction,  A  increases  with  temperature,  U  decreases  (Type  II. 
Fig.  89).  For  further  illustrations,  cf.  Nernst  (Sitzungsber. 
Berlin.  Akad.,  Jan.,  1909;  Journal  der  Physique,  l.c.,  1910), 
and  Pollitzer,  Ahrens.  Sammlung).  The  following  table  con- 
tains the  values  of  the  e.m.f.  of  certain  cells  consisting  of 
"pure"  solid  or  liquid  components  first  calculated  by  the 
aid  of  the  theorem  and  secondly  observed  directly. 


Chemical  reaction. 

T. 

E  calculated. 

E  observed. 

Pb  -f  2AgCl     ->  2Ag  +  PbCl, 

200 

0-4890 

0-4891 

2Hg  +  2AgCl  ->  Hg2Cl2  +  2Ag 

288 

0-0437 

0-0439 

Pb  +  I2             ->  PbI2 

291 

0-863 

0-863 

2Ag+I2             -»2AgI 

291 

0-618 

0-678 

Further  work  on  similar  types  of  cells  has  been  carried  out, 
in  which  not  only  is  use  made  of  the  Nernst  Theorem  but 
likewise  of  the  Einstein  theory  of  atomic  heat  of  solids  (as 
modified  by  Nernst  and  Lindemann).  This  will  be  taken  up 
after  we  have  discussed  Planck's  Unitary  Theory  of  radiant 
energy  and  Einstein's  application  of  it  to  the  heat  capacity  of 
solids  (Part  III.,  Chap.  II.). 


376       A   SYSTEM  *OF  PHYSICAL   CHEMISTRY 


APPLICATION    OF    NERNST'S    THEOREM    TO    EQUILIBRIA    IN 
HOMOGENEOUS  GASEOUS  SYSTEMS  OR  DILUTE  SOLUTIONS. 

Although  as  already  pointed  out  in  several  instances  the 
theorem  directly  applies  only  to  solid  or  liquid  systems,  it 
will  be  shown  that  it  can  also  be  extended  to  the  calculation 
of  the  equilibrium  constant  K  in  a  gaseous  system,  pro- 
vided we  know  beforehand  the  heat  of  the  reaction  at  a 
single  temperature,  the  molecular  heats  of  the  gaseous  sub- 
stances at  a  few  temperatures,  and  the  integration  constants  of 
the  integrated  form  of  the  Clapeyron  vapour  pressure  ex- 
pression, which  has  been  discussed  in  an  earlier  part  of  this 
book.  Suppose  the  reaction  under  discussion  is  the  general 
one  — 


If  this  reaction  were  taking  place  in  a  completely  con- 
densed system,  all  the  substances  being  single  "  pure  "  solids 
or  liquids  (liquid  solutions  being  ruled  out  since  at  low 
temperatures  these  would  probably  separate  into  their  single 
components),  we  could  represent  the  total  energy  change 
(i.e.  decrease)  by  the  usual  expression— 

U  =  U0  +  aT  +  J3T2  +  yT3  +,  etc. 

If  the  same  reaction  were  made  to  take  place  in  the  homo- 
geneous gaseous  state,  we  can  express  the  total  energy  change 
(for  the  same  number  of  molecules  as  before)  by  the  term  U', 
where  — 

U'  =  U'0  +  a'T 


U'o  represents  the  heat  evolved  by  the  reaction  in  the 
neighbourhood  of  the  absolute  zero.  At  the  actual  zero  the 
existence  of  a  gaseous  state  is  supposed  to  be  impossible. 
Now  we  have  already  seen  that  the  affinity  of  a  reaction  such 
as  the  above  occurring  in  a  condensed  system  (liquid  or  solid) 
is  given  by  a  particular  form  of  the  van  't  Hoff  Isotherm,  viz., 

A  =  RT  log  K  —  RTJEv  log  C, 


where  E 


NERNSTS  HEAT  THEOREM  377 


icre  K  is  the  equilibrium  constant,  namely  : 

["Product  of  concentration  of  resultants  raised  to  required  powers'! 
[.Product  of  concentration  of  reactants  raised  to  required  powersj 

for  the  reaction  in  the  dilute  gaseous  state  at  the  same  temperature 
T;  and  C  denotes  the  concentration  of  saturated  vapour  of 
each  constituent  separately.  The  term  Sv  log  C  has  the 
significance  — 

v\  log  Cc  +  v'z  log  Cd  .  .  .  —  vi  log  Ca  —  v2  log  C6 

Further  we  have  already  seen  (p.  96)  that  the  integration 
of  the  Clapeyron  vapour  pressure  formula  leads  to  values  of  C, 
the  concentration  of  saturated  vapour  which  can  be  expressed 
for  a  single  substance  by  the  relation  — 


...  +  ;,     (6) 

in  which  "  i  "  is  a  characteristic  constant  for  the  substance 
The  terms  AO,  a0,  j30,  y0  can  be  calculated  from  the  relation 
(Am  —  RT)  =  the  molecular  internal  latent  heat  of  vaporization, 


where  AQ  denotes  the  internal  molecular  heat  near  absolute 
zero.  It  follows  from  the  definition  of  the  term  ZV  log  C  that 
we  can  write  — 


»*C-^  +  ^  +  -=^-*  + 

z^V^T+^+,et,  +  ^ 

+"iV    ^a,,,,  log  T    I.A.T 

"XT'       ~R R ' etc'     1  - 


+yzAo6     v2a06  log  T     y^T 
"^T~  "^T  R     ~' 


or  more  briefly — 
Sv  log  C 


3?8        A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

Employing  the  van  't  Hoff  isotherm  in  the  form  given 
above,  we  obtain  — 


RT  log  K  =  RT^  log  C  +  A 

Now  the  affinity  A  of  a  reaction  occurring  in  a  condensed 
system  is  given  by  Nernst's  Theorem  —  note  at  this  point  the 
theorem  is  introduced  —  viz.— 


=  U0  —  j8T«  —     T 


Hence  we  can  write  after  substitution,  using  equation  (6A) 


RT  log  K  =  —  SV\Q  +  27vooT  log  T  +  27i>ft>T2  +  - 

2 


U0  -  j3T2  - 
That  is,  RT  log  K=(U0—  2v\^)+SvaQT  log  T—  (j8— 

...     (7) 


Equation  (7)  may  be  regarded  as  the  expression  for  the 
equilibrium  constant  of  a  dilute  gaseous  reaction  obtained 
by  the  application  of  the  Nernst  Theorem  to  the  van  't  Hoff 
isotherm  for  the  same  reaction,  when  carried  out  in  the  liquid 
or  solid  state. 

Now  we  have  already  seen,  in  the  earlier  part  of  this  book, 
in  dealing  with  the  problem  of  the  effect  of  temperature  on  the 
equilibrium  constant  of  a  gaseous  reacting  system,  that  such  is 
given  by  the  reaction  isochore,  viz.  — 

-TJ'  __  d  log  K 
RT2  ~~      <9T 

This  equation  on  integration  (see  below)  contains  an 
integration  constant  I,  to  the  evaluation  of  which  thermo- 
dynamical  considerations  (ist  and  2nd  Laws)  have  nothing 
to  say.  Hitherto  the  only  way  of  obtaining  I  was  by  actually 
measuring  K  for  at  least  one  temperature.  Now,  however,  — 
and  in  this  lies  the  great  advance  made  by  the  application  of 
the  Nernst  Theorem  to  the  calculation  of  K  —  we  can  find  the 


NERNSrS   HEAT  THEOREM  379 

real  (physical)  meaning  of  I  and  can  show  how  the  I  of  a  reaction 
may  be  calculated  beforehand  quite  apart  from  the  reaction  itself  ^ 
if  we  only  know  the  vapour-pressure-temper  atiire  relations  of  each 
of  the  substances.  This  is  effected  as  follows  :  — 

We  have  seen  in  an  earlier  section  (p.  149)  that  the 
integrated  form  of  the  isochore  (quite  independent  of  Nernst's 
Theorem,  of  course)  for  the  gaseous  reaction  may  be  written  — 

RTlogK=UV-a'TlogTH3'T2-^T3-,  etc.+RTI  (8) 

The  significance  of  the  coefficients  is  evident  from  the 
equation  — 

U'  =  U'0  +  a'T 


U'  being  the  heat  evolved  at  constant  volume  or  decrease  in 
total  energy,  in  the  reaction  in  the  homogeneous  gaseous 
state.  Now  equations  (7)  and  (8)  are  both  expressions  for 
the  same  quantity  RTlogK.  They  must  be  identical. 
Equating  the  coefficients  of  like  terms,  we  find  that  — 


—a  = 


-/  =  —  7 

and  I  ==  Evi 

The  important  relation  for  our  present  purpose  is  the  last 
one.  It  shows  that  (on  the  basis  of  Nernst's  Theorem)  the 
sum  of  the  integration  constants  of  the  vapour  pressure  curves 
which  can  be  directly  determined  may  be  used  to  calculate  the 
constant  I  for  a  given  gaseous  reaction,  without  actually  carry- 
ing the  reaction  out  at  all.  We  can  thus  rewrite  the  integrated 
form  of  the  reaction  isochore  (viz.  equation  (8))  in  the  form  — 

RT  log  K  =  U'0  —  a'T  log  T  —  )3'T2  —  y-  T3  +  RT27w 
or         logK=-logT-T-y-T2  +  ^/    (9) 


380       A    SYSTEM  *OF  PHYSICAL   CHEMISTRY 

For  practical  convenience  we  can  convert  natural  logarithms 
into  Iog10,  and  the  expression  becomes  — 


K  denotes  a  ratio  of  concentration  terms,  namely,  — 


For  greater  convenience  it  is  sometimes  advisable  to  work 
in  terms  of  partial  pressures.  The  equilibrium  constant  is 
now  Kp,  where  — 


Assuming  the  applicability  of  the  gas  law  for  each  of  the 
constituents,  we  can  write  p  =  RTC,  and  hence  — 


= 


Hence  Iog10  Kp  =  Iog10  K  +  Ev  Iog10  R  +  Sv  Iog10  T 
So  that  equation  (QA)  may  be  written  — 

(%-^R)  pr       y 

-  log*  1   -  -      - 


2-3023 

The  expression  —  ^—  —  —  °g        is  denoted  by  2vCr\,  where 
2-3023 

CQ  is  the  chemical  constant  of  any  particular  substance,  values 
of  which,  as  we  have  already  seen,  are  given  in  an  earlier  part 
of  this  book  (p.  97).  On  making  this  slight  substitution,  we 
obtain  finally  for  the  integration  of  the  isochore  with  the  help 
of  Nernst's  Theorem  — 


9-14 


NERNST^  HEAT  THEOREM        381 

Equations  (9),  (QA),  (QB),  and  (90)  which  follows,  are  of 
course  all  equivalent.  A  few  words  more  with  regard  to  the 
numerical  evaluation  of  the  coefficients  which  occur  in  the 
above. 

We  have  seen  that  a!  =  —  EVOLQ,  where  a0  is  obtained 
directly  from  the  vapour  pressure  curve  of  each  substance 
(equation  (6)),  from  which  C0  is  also  obtained.  The  expression 

a'  —  Zv  R  =  —  ZvaQ  —  ZVR  =  —  2v(aQ  +  R) 

Nernst  puts  (a0  +  R)  =  3*5  (calories)  as  an  approximation 
(which  is  probably  only  a  rough  one,  cf.  Haber,  Thermo- 
dynamics of  Technical  Gas  Reactions  >  appendix  to  Lecture  3  ; 
also  Nernst,  Applications  of  Thermodynamics  to  Chemistry  ', 
p.  77).  Hence 


(q'-2frR)  loglo  T  =  -ZV  1  75  loglo  T 
R 

when  R  has  been  taken  to  be  2  calories.     Equation  (96)  can 
therefore  be  written  — 

Iog10  Kp 

-j5sL-h»  I-75log1oT-4-^T-9-^T2+^C0    (9c) 

Knowing  the  molecular  heats  of  the  gaseous  reactants  and 
resultants  at  a  few  temperatures,  we  can  calculate  a',  /?',  and 
y',  since  — 

Molecular  heat   capacity   oft      ^, 

reactants—  Molecular  heat|==  ^=a'-f2j8'T-f  3y'T2+.  .  . 
capacity  of  resultants 

Knowing  these  coefficients,  and  likewise  knowing  the  total 
energy  change  (for  the  gaseous  reaction)  U'  at  a  given  tem- 
perature T,  we  can  calculate  U'0  by  means  of  the  familiar 
relation  U'  =  U'0  +  a'T  +  j8'T2  +  y'T3  ...  As  a  matter  of 
fact,  the  factor  •/  may  be  neglected  in  general,  and  in  some 
cases  even  j3'.  Referring  again  to  equation  (90),  it  should 
be  noted  that  for  reactions  (gaseous)  in  which  there  is  no 
change  in  the  number  of  molecules  (e.g.  H2  -f-  C12  =  2HC1) 
the  term  2?vi'75  log  10T  necessarily  becomes  zero.  The  terms 


382       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 


0  as  well  as  ]8'  are  not,  however,  necessarily  zero  for  this 
or  any  other  type  of  reaction. 

It  must  be  clearly  borne  in  mind  that  the  vital  point  of  the 
problem  under  discussion  lies  in  the  possibility  of  calculating 
the  isochore  integration  constant  I  from  the  characteristic  Zvi 
or  the  chemical  constants  2vC0  of  the  substances  taking  part. 
This,  as  already  pointed  out,  is  the  conclusion  arrived  at  by 
applying  the  Nernst  Theorem  to  reactions  in  the  gaseous 
state.  Nernst,  himself,  has  pointed  out  (Applications  of 
Thermodynamics  to  Chemistry  ',  p.  55)  that  Le  Chatelier,  even 
as  early  as  1888,  had  appreciated  the  significance  of  the 
constant  I,  and  "seems  also  to  have  had  some  idea  of  the 
method  of  its  solution."  The  following  are  Le  Chatelier's 
words  : 

"It  is  very  probable  that  the  constant  of  integration  [I], 
like  the  coefficients  of  the  differential  equation  [the  Gibbs- 
Helmholtz  equation]  is  a  definite  function  of  certain  physical 
properties  of  the  reacting  substances.  The  determination  of 
the  nature  of  this  function  would  lead  to  a  complete  know- 
ledge of  the  laws  of  equilibrium  ;  it  would  make  it  possible  to 
determine,  a  priori,  all  the  conditions  of  equilibrium  relating 
to  a  given  chemical  reaction  without  the  addition  of  new 
experimental  data." 

EXAMPLES  OF  THE   APPLICABILITY  OF  NERNST'S   THEOREM 
TO  HOMOGENEOUS  CHEMICAL  EQUILIBRIA. 

i.  The  Dissociation  of  Water  Vapour. 
The  reaction  is  2H2O  ->  2H2  +  °2- 


these   partial   pressure   terms  being   equilibrium  values.      If 
x  is   the   degree   of  dissociation  under   i   atmosphere  total 

pressure,  then  — 

*3 

K*~  2 
(since  the  degree  of  dissociation  is  very  small).     Experiment 


HEAT   THEOREM  383 

has  shown  that  at   1300°  absolute,  and  under  i  atmosphere 
pressure,  x  =  0*29  X  io~"4 

.-.  Kp  =  1-218  X  lo-i4 
and  Iog10  Kp  =  —  13-91 

We  shall  now  proceed  to  calculate  Iog10  Kp  from  equation 
(90).  Experimental  data  gives  us  the  following  — 

U'0  =  —114,500  calories  per  2  moles  of  water  vapour 

[N.B.  —  The  negative  sign  denotes  an  increase  in  total 
energy  or  absorption  of  heat.  This  is  what  one  would  expect 
on  the  Le  Chatelier  principle  for  the  dissociation  of  water, 
since  the  extent  of  the  dissociation  is  known  to  increase  as 
the  temperature  rises.] 

Further  |8'  =  —  0-00128 

/  =  4-  67  X  io-7 

Sv  =  (2  moles  of  H2  +  i  mole  of  O2  —  2  moles  H2O)  =  i 

•*•  2"  i-75  Jogio  T  =  1-75  iQgio  T. 
The  chemical  constant  C0  per  mole  of  H2  =  2-2 
/.  for  2  moles,  2C0  =  4-4 
C0  per  mole  of  O2  =  2-8 
C0  per  mole  of  H2O  (gas)  =  37 
/.  for  2  moles,  2C0  =  7-4 
.'.  27vC0  =  4-4  +  2-8  —  7-4  =  —  0-2. 

Equation  (90)  becomes  therefore  — 


= 
' 


4'57i  4-571  9-14 

or    Iog10  Kp  =  "~2^°50  +  1-75  Iog10  T  +  0-00028T  —  0-2 

neglecting  the  •/  term. 

For  the  temperature  1300°  absolute  this  equation  gives  the 
value  — 


=  —  14-00 

While  Iog10  Kp  observed  =  —  1  3*9  1 

The  agreement  is  very  satisfactory.     It  should  be  pointed 


384       A   SYSTEM  &F  PHYSICAL   CHEMISTRY 

out  that  the  last  three  terms  on  the  right  hand  side  are  often 
small  compared  with  the  first  two.  This  is  a  pity,  since  it  is 
just  the  final  term  which  particularly  interests  us. 

2.  The  Deacon  Process  of  Chlorine  Manufactured 

The  reaction  is— 

4HC1  +  O2  =  2H2O  +  2C12 

This  reaction  may  theoretically  be  split  up  into  the 
simpler  reactions — 

2H2O  =  2H2  +  O2 
2HC1  =  H2  +  C12 

For  the  first  reaction— 


two 


(these  partial  pressures  being,  of  course,  equilibrium  terms). 
It  has  just  been  shown  that  — 

+  175   log  T  -  0-00028T  -  0-2 


For  the  second  reaction  — 


and  it  has  been  found  that  — 


In  this  the  j3'  as  well  as  the  y'  term  is  omitted  as  negligible. 
Also,  since  there  is  no  change  in  the  number  of  molecules,  the 
expression  Zi>  175  Iog10  T  =  o  (though  of  course  2vCQ  is 
not  zero). 

For  the  Deacon  Process  itself  we  have  — 


TT  __,2 

JS.Jp  Deacon  —  ^~~^'^ 
/HC1  XA>2 

1  Cf.  Vogel  v.  Falkenstein,  Zeitsch.  Elektrochemie,  12,  763,  1906. 


NERNSrS  HEAT   THEOREM 


385 


(these  partial  pressure  terms  being  equilibrium  values).     This 
may  be  rewritten  — 

/H20  X/Cl, 


that  is— 


K2 

Deacon  = 


.*.  Iog10  Kp  Deacon  =  2  Iog10  K2  —  log 

Using  the  above  data,  one  finds  — 

1°§10  Kp  Deacon 


or 


<-p  Deacon  — 


+  579° 


-1*75  logT+o'ooo28T— 0*2 

t 

-  1*75  log  T  —  0-00028T  —  1-4 


The    following   table,  given    by   Nernst   (Applications  of 
Thermodynamics  to   Chemistry^  p.  89),  shows  the  agreement 
between  the  values  thus  calculated  and  those  experimentally 
determined  in  an  exceedingly  accurate  manner  by  Vogel  v. 
Falkenstein. 


,o  c           *S>  Deacon 
observed. 

Kp  Deacon 

calculated. 

log  K^,  Deacon 
observed. 

l°Z  Kp  Deacon 
calculated. 

I 

450 
600 
650 

3I-0 
0-893 
0-398 

3^9 
0-98 
0-371 

—  0-050 
—  0-400 

1*50 
—  O*OO9 
-0*430 

3.  The.  Dissociation  of  Iodine  Vapour  ? 
The  reaction  is  — 


Exceedingly  accurate  measurements  of  the  equilibrium 
constant  at  various  temperatures  ranging  from  800°  C.  to 
1200°  C.  were  made,  the  following  being  the  observed  values 
of  Kp  in  partial  pressure  terms. 

800°  C.       900°  C.      1000°  C.      1100°  C.     1200°  C. 

Kp  =  0*0114        0-0474        0*165          0*492          1*23 

1  G.  Slarck  and  Max  Bodenstcin,  Zcit.  Elektrochem.^  16,  961,  1910. 
T.P.C.  —  II.  2  C 


386       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

C2 

Measurements  of  K  =  -^  were  also  carried  out,  and  from 

*"lj 

these  by  employing  the  isochore  equation  in  its  unintegrated 
form  over  small  temperature  ranges,  in  which  U'  was  con- 
sidered to  remain  constant,  viz.  — 

dlogK_  —  U' 
dT        ~  RT2 

values  of  U'  were  obtained. 

Equation  (9^)  is  written  by  Bodenstein  after  substitution  of 
numerical  values  thus  — 


.75  log10T  - 


The  term  EvCQ  is,  according  to  Bodenstein,  -j-  0*422. 
Employing  this  equation  one  can  calculate  Iog10  Kp. 

The  following  table  shows  the  very  good  agreement  ob- 
tained between  observed  and  calculated  values. 


Absolute  temperature. 

1073° 

"73° 

-  i'325 
-  i'34 

1273° 

1373° 

1473° 

Log10  Kjp  observed 
Logi0  Kp  calculated     . 

-  !'945 
-  1-956 

-0782 

-o77. 

-  0-309 
-  0-3II 

—  0-09I 

—  0*084 

Note,  on  the  calculation  involved  in  the  above. — For  the 
actual  calculation  of  U'0  it  is  more  convenient  to  deal  with 
the  heat  of  reaction  (at  any  temperature  T)  at  constant  pres- 
sure, viz.  Qp.  This  quantity  is  simply  (U'  -\-  RT),  since  in 
the  above  case  2  moles  of  I  (atomic  iodine)  are  formed  from 
i  mole  of  I2,  the  external  work  being  therefore  RT.  The  ex- 
pression Q'p  can  be  expressed  as  Q'0  or  U'0  -f-  a  series  of 
ascending  powers  of  T,  each  coefficient  being  determined  from 
the  molecular  heats  at  constant  pressure  of  the  reactants  and 
resultants,  and  hence  U'0  can  be  calculated.  On  our  notation 
we  have  written  for  a  gaseous  reaction — 

U'  =  U'0  +  o/T  +  j3'T2. 


rS   HEAT   THEOREM  387 

:e  from  the  equation  Q'p  =  U'  +  27vRT  we  get— 

RT=(a'+27vR)T+j3'T2+,  etc. 
and  ^  =  a'  +  27vR  +  2j3'T  +,  etc. 

Now  the  molecular  heat  at  constant  pressure  of  any 
monatomic  gas  (I)  at  any  temperature  is  5  calories  per  degree. 
Also  the  molecular  heat  of  iodine  (molecular)  I2,  is  shown  by 
experiment  to  be  given  by  the  expression — 

Hp  =  6-5  +  o-oo38T 

Hence  the  molecular  heat  at  constant  pressure  of  reactants 
-  molecular  heat  of  resultants  which  we  may  denote  by 

2vUp  =  (6-5  +  0-00381)  —  2X5  =  —3*5  +  o-oo38t 
Near  zero  the  value  of  2vHp  —  a!  +  EvIL 
But  in  the  neighbourhood  of  absolute  zero— 

SvUp  =  6-5  —  10  =  —3-5 

.-.  a'  +  ZVR  =  —  3-5 

Now  we  have — 

2vHp  =d-^}  =  a'  +  ZVR  +  3j8'T 

•'.  -3'5  +  o-oo38T  =  -3'5  +  ajS'T 
.*.  j3'  =  0*0019 

It  should  be  mentioned  that  the  value  employed  for  2vCQ 
is  one  which  happens  to  fit  the  equation  (at  least  at  one 
temperature).  The  total  expression,  however,  as  is  seen  from 
the  above  table,  holds  over  the  entire  range  investigated  and 
in  all  probability  over  any  range  whatsoever  (as  long  as  the 
term  containing  y'  can  be  neglected).  Having  thus  obtained 
the  information  that  -£VC0,  that  is  2  X  C0l—  C0l<>  =  +0*422, 
Bodenstein  has  calculated  the  value  of  C0l,  using  the  known 
value  +  4*0  for  C0l  .  C0l  thus  comes  out  to  be  2*2. 
Bodenstein  points  out  that  this  is  the  first  instance  in  which 
the  chemical  constant  for  a  monatomic  substance  has  been 
accurately  obtained. 


388       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

Modification  of  the  Equation  for  Homogeneous  Gaseous 
Equilibria. 

As  we  have  seen,  the  terms  containing  j3'  and  y'  can  be  in 
general  neglected.  Also  the  value  of  U'0  is  very  nearly  equal 
to  the  heat  developed  at  ordinary  temperatures  and  under 
constant  pressure  (this  being  the  quantity  for  which  experi- 
mental data  can  most  conveniently  be  obtained).1  Equation 
(90)  can  therefore  be  written  in  the  following  approximate, 
but  at  the  same  time  useful,  form  — 


logic  Kp  =   .          +  2»i  -7  5  log  M  T  +  ZvC 


For  examples  of  the   use   of  this   formula,  cf.    O.    Brill, 
Zeitsch.  physik.  Chem.,  57,  721,  1907. 

APPLICATION  OF  NERNST'S  THEOREM  TO  THE  CALCULATION 
OF  EQUILIBRIA  IN  HETEROGENEOUS  SYSTEMS  (GAS  — 
SOLID  OR  GAS  —  LIQUID). 

Let  us  take  as  a  typical  instance  the  dissociation  of  calcium 
carbonate,  viz.  — 

CaC03  =  CaO  +  CO2 

The  isotherm  for  the  affinity  A  of  the  above  reaction  when 
all  components  are  in  the  solid  state  is  — 

A  =  RT  log  K  —  RTZV  log  C 
where  K  =  C€Ca°X  Cpc°2,  and  C  in  the  last  term  refers  to 


saturation  concentration  of  the  vapours. 

The  two  terms  Cecao  and  Ctcaco3  are,  however,  simply 
constant  concentrations,  since  the  lime  and  calcium  carbonate 
are  present  in  the  solid  form.  The  isotherm  may  thus  be 
written  — 

A  =  RT  log  C6C0.5  —  RT  log  Csaturated  C02 

1  The  value  of  U'0  used  for  the  2HC1->H8  +  CL  reaction  is  really 
this  quantity. 


NERNSTS  HEAT   THEOREM 


389 


The  term  Evi  in  the  vapour  pressure  expression  (equa- 
tion (6A))  becomes  simply  i  for  CO2  alone,  and  the  equation 
becomes  identical  with  equation  (6)  itself.  Hence  in  the  final 
expression  (equations  (9),  (9A),  (QB),  or  (90))  the  expression 
ZVC0  is  simply  the  value  of  C0  for  i  mole  of  CO2  gas.  This 
is  taken  by  Brill  (loc.  tit.)  to  be  3*2.  The  term  log  K^ 
becomes  identical  with  log  /,  where  p  is  the  equilibrium  pres- 
sure of  CO2  in  the  presence  of  CaO  and  CaCO3.  For  such 
heterogeneous  equilibrium  the  approximate  form  of  equa- 
tion (90)  may  be  used,  viz.  — 


log10T 


4 


On  substituting  the  equivalents  for  these  terms  given  above, 
one  finds  finally  for  the  dissociation  equilibrium  of  any  carbonate 
the  expression  — 

loglo  K,  =   .9'?    +  1-75  log  T  +  3-2 


Q'p  denotes  the  heat  of  dissociation  of  the  carbonate  in- 
vestigated. Since  such  dissociation  increases  with  increasing 
temperature,  the  term  Q'p  is  heat  absorbed  and  must  therefore 
be  taken  with  a  negative  sign.  The  following  table  (cf.  Brill, 
/.*-.,  p.  736)  contains  the  values  of  Tj  (calculated  and  observed) 
when  the  dissociation  pressure  /  =  i  atmosphere.  In  this 
case  logjo  p  =  o,  and  hence  — 


4' 


Substance,    j  Q'p  observed  by  Thomsen.      xx  observed. 


AgCO3 

—  20,060 

498° 

PbC03 

-22,580 

575° 

MnCO3 

i     -23,500  (Berthelot) 

608° 

CaC03 

-42,520 

1098° 

SrCO3 

-55.770 

1428° 

T  calculated. 

Observer. 

i 

548° 
6  10° 

Joulin 
Colson 

632° 
1091° 
1403° 

Joulin 
Brill 
Brill 

Several  other  illustrations  of  the  applicability  of  the  theorem 
to  heterogeneous  systems  in  which  gases  are  present  are  cited 


390       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

by  Nernst  (Applications  of  Thermodynamics  to  Cfamistry^ 
p.  96,  seq.).  In  the  present  instance  we  shall  discuss  only  one 
further  problem,  which  is  of  great  importance,  namely,  the 
calculation  of  the  electromotive  force  of  a  gas  cell  from  purely 
thermal  data.  To  apply  the  theorem  to  such  a  system,  which 
is  not  an  entirely  condensed  one,  we  have  to  proceed  in 
principle  as  follows  (quotation  from  Nernst,  Sitzungsber.  Berliner 
Akad.,  p.  255,  1909):— "We  think  of  the  cell  at  a  tempera- 
ture so  low  that  the  gases  become  either  liquid  or  solid,  and 
apply  our  theorem  to  this  state.  We  then  take  into  consider- 
ation the  equilibrium  between  the  solid  or  liquid  substances 
and  their  vapours  and  thence  by  applying  the  Second  Law  of 
Thermodynamics  we  can  carry  out  calculations  for  any  tem- 
perature and  pressure."  In  this  way  we  can  realise  the  con- 
ditions obtaining  in  an  actual  gas  cell.  A  typical  illustration 
is  the  Knallgas  cell,  in  which  the  reaction — 

2H2  +  O2->2H2O 

gas  gas  liquid 

goes  on  and  yields  the  electromotive  force.  In  this  reaction 
the  equilibrium  constant  in  the  homogeneous  phase  is  given 

by- 
pa 

•K:  -  ^eHaO 

r3 — v~r~ 

^eH2  A  ^e02 

If  the  reaction  were  made  to  take  place  under  such  circum- 
stances that  all  the  substances  were  pure  liquids  or  solids,  the 
affinity  A  of  the  reaction  is  given  by  (i.e.  the  affinity  of  forma- 
tion of  2  moles  of  H2O)— 

A  =  RT  log  K  -  RTZV  log  CBaturated 

Now  exactly  as  in  the  case  of  the  dissociation  of  the 
carbonates,  the  term  for  the  concentration  of  saturated  water 
vapour  present  in  both  of  the  above  terms  cancels.  The  term 
2v  log  C  may  therefore  be  taken  as  equivalent  to— 

-(2logCHa    +    logC02) 

Saturated  vapour  Saturated  vapour 

over  liquid  hydrogen         over  liquid  oxygen 

[N.B.— The  negative  sign  comes  into  the  above  expression 
since  the  terms  for  H2  and  O2  occur  in  the  denominator  of 


NERNSrS   HEAT   THEOREM  391 

the  original  expression  for  K.]  Passing  to  partial  pressure 
terms,  and  still  omitting  the  term  for  water  vapour,  we  see 
that— 

log  K^  =  log  (   2       -*—  —  )  =  -(2  log  An2    +    log  Ao2) 


X 

Equilibrium      Equilibrium  Equilibrium  Equilibrium 

pressure  pressure  pressure  pressure 

Proceeding  as  before,  we  reach  equations  (9),  (g\),  (96)  or 
(90).  In  these  2vCQ  ={C0  for  resultants  —  C0  for  reactants}; 
and  as  the  resultant  H2O  term  is  dropped,  since  it  occurs 
in  both  terms  in  the  expression  for  A,  it  follows  that  — 

EvCQ  =  -(2COH2  +  C002)  =  ~7'2 

One  should  observe  that  the  term  Sv  in  the  above  case 
has  the  value  —  3,  since  it  must  always  represent  (number  of 
molecules  of  resultants  —  number  of  molecules  of  reactants) 
which  becomes  in  this  case  simply  the  number  of  molecules  of 
reactants  taking  part,  viz.  (iH2+Oj).  The  term  27vi*75  Iog10  T 
in  equation  (90)  becomes  now  —  3  X  1*75  Iog10  T,  or 
—  5'25log10T.  The  term  U'0,  i.e.  total  energy  change  in 
formation  of  2  moles  of  liquid  water  in  this  case,  has 
been  found,  from  direct  measurements  of  heat  of  forma- 
tion of  liquid  water  from  the  gases  hydrogen  and  oxygen  at 
constant  pressure  at  several  temperatures,  to  be  137,400 
calories.  In  the  formation  of  water,  heat  is  evolved,  that  is, 
the  total  energy  decreases  and  hence  U'0  appears  with  a 
positive  sign.  Equation  (90)  can  thus  be  written  — 


Ao2 
logio  ^  =™       -  5^5  logio  T  +  o-oioT  - 


from  which  Iog10  Kp  can  be  calculated. 

Now  the  affinity  A  of  the  reaction  2H2  4-  O2->  2H2O,  in 
which  the  reactants  are  gaseous  and  the  resultant  a  liquid,  i.e. 
the  affinity  of  the  actual  process  occurring  in  the  Knallgas  cell, 
is  given  by— 

A  =  RT  log    s  —  RT  log  -3- 


392       A   SYSTEM   OF  PHYSICAL   CHEMISTRY 

where  the  term  for/H2o  has  been  eliminated.  If  the  oxygen 
and  hydrogen  are  supplied  to  the  cell,  both  at  i  atmosphere 
pressure,  the  above  expression  reduces  to — 

A  =  RT  lQg     — ^ RT  log  i  =  RT  log  - 

xAo2 


This  can  be  calculated  numerically  from  equation  (90) 
given  above.  We  can  thus  calculate  the  affinity  of  the  re- 
action in  the  Knallgas  cell.  Now  A  refers  to  the  formation 
of  2  moles  of  water.  To  form  2  moles  of  water  electrically 
requires  four  faradays,  and  hence  A  =  4E,  where  E  is  the 
electromotive  force  of  the  cell.  Calculating  Kp  as  above  for 
the  temperature  290°  absolute,  and  obtaining  the  numerical 
value  of  A  from  this,  and  hence  the  numerical  value  of  •  E, 
one  obtains  from  purely  thermal  data>  by  the  aid  of  Nernsfs 
Theorem,  that— 

E29o°abs.  =  1*242  volts 

We  have  now  to  compare  this  value  with  the  observed. 
Direct  observation  of  the  electromotive  force  of  this  cell  has 
shown  £290°  aba.  to  be  1-15  volts,  but  as  already  pointed  out 
this  is  certainly  too  low  owing  to  the  difficulty  of  saturating 
the  platinum  electrode  with  oxygen.  On  applying  the  principle 
of  virtual  work,  the  expression  obtained  for  the  e.m.f.  of  this 
cell  is— 

RT  i 

where  7THg  and  7r02  denote  the  partial  pressures  of  hydrogen 
and  oxygen  at  the  temperature  T  in  saturated  water  vapour.1 

1  In  the  cell  H2  |  electrolyte  |  O2,  if  /Ha  and/02  represent  the  (arbi- 
trarily chosen)  partial  pressures  of  the  gas  electrodes,  and  /H20  is  the 
vapour  pressure  of  undissociated  water,  then  if  the  formation  of  2  moles 
of  water  took  place  as  vapour  only,  the  work  A  would  be — 
A  =  RT  log  Kp  -  RT2J/  log/ 

=  RT  log  -^%^ RT  log 


Since  in  the  presence  of  liquid  water  /eH20  =  /H20  we  can  w"t 
A  =  RT  log    ,      '  -  RT  log    2     ' 


Also  if  we  use  the  symbols  TT^   f°r/ejj    an^  TQa  for/eO  ,  and  also  feed  in 


HEAT   THEOREM  393 

At  fairly  high  temperatures  the  degree  of  dissociation  of  water 
vapour  has  been  experimentally  obtained,  and  by  simply  using 
the  isochore  expression  in  its  integrated  form  containing  the 
constant  I,  which  is  determined  from  a  directly  obtained  value 
of  K^  (no  use  being  made  of  the  Nernst  Theorem),  one  finds 
that  at  290°  abs.  the  value  of  — 

TTH,  =  0-0191  X  i'8o  X  io~27  atmospheres 

0-0191  X  r8o  X  io~27 
and         7r0a  =  —  —  „ 

whence  E29o°abs.=  1*232  volts  (Nernst  and  von  Wartenberg, 
Zeitsch.  physik.  Chern^  56,  544,  1906).  Other  values  are 
E  =  1-224  (G.  N.  Lewis,  Zeitsch.  physik.  Chem.t  55,  449,  1906)  ; 
and  E  =  1*234  (Bronsted,  Zeitsch.  physik.  Chem^  65,  744,  1909). 
This  may  be  regarded  as  the  true  "  observed  "  value  for  the 
electromotive  force  of  the  Knallgas  cell,  and  it  is  clear  that 
this  agrees  pretty  well  with  that  calculated  by  the  aid  of  the 
Nernst  Theorem,  viz.  E2go°  =  1*242  volts. 

The  discrepancy  which  does  exist  is  discussed  by  Nernst 
(I.e.).  It  is  due  to  the  abnormal  behaviour  of  liquid  water  in 
respect  of  its  molecular  heat.  By  working  with  the  data  for 
ice,  Nernst  finds  that  at  o°  (i.e.  the  temperature  at  which 
water  and  ice  are  in  equilibrium  under  i  atmosphere  pressure), 
the  electromotive  force  of  the  cell  calculated  from  purely 
thermal  data  should  be  1*2393  volts.  To  find  the  value 
of  this  at  290°  abs.  we  can  simply  use  the  Gibbs-Helmholtz 
equation  directly,  viz. 

Q  <*E 


whence  -j-  =  —  0*00085 


the  gases  at  the  electrodes  at  i  atmosphere  pressure,  then 
A  =  RTlog 

and  since  the  formation  of  two  moles  of  water  from  the  ionic  state  would 
require  4  unit  charges  (faradays),  it  follows  that — 

~       A      RT  .  i 

E  =  —  =  —  log  -^ . 

4          4  ^rio  X  TTO 


394       A   SYSTEM  OF  PHYSICAL    CHEMISTRY 

Using  this  value  for  — ,  one  finds  that— 
at 

£290°  =  1*225  volts 

a  quantity  which  agrees  very  well  with  the  "  observed  "  value. 
The  Nernst  Theorem  can  thus  be  satisfactorily  applied  to 
the  reactions  taking  place  in  gas  cells.1     The  following  table 
contains  the  values  obtained  with  a  few  gas  cells. 


Reaction. 


Temperature.        K  observed.          K  calculated. 


2lI2+O2->2lL,O       .....' 

2Ag  +  C12  ->  2AgCl  .     .     . 

290° 
290° 

1-232 
1-157 

I--225 
I-092 

Pb  +  Cl2->PbCl2       .     .     . 

290° 

1-612 

If594 

\    solution  ' 

3030 

ri6o 

1-170 

H  +  Cl  —  >  2HC1  I1  normal 
\    solution 

298° 

1-366 

1-365 

We  have  already  referred  to  the  somewhat  analogous 
problem  of  the  electromotive  force  of  cells  in  which  the 
reaction  takes  place  in  solution.  This  is  to  be  distinguished 
from  reactions  occurring  between  pure  condensed  substances 
(i.e.  each  phase  consisting  of  a  single  chemical  entity).  To 
the  latter  we  have  already  seen  that  the  theorem  of  Nernst 
can  be  most  directly  applied.  For  the  case  of  solutions  we 
can  proceed  by  the  help  of  the  artifice  already  exemplified  in 
the  case  of  the  Clark  cell,  i.e.  calculating  the  electromotive 
force  at  such  temperatures  that  solid  phases  are  present, 
and  then  with  the  aid  of  subsidiary  data,  e.g.  solubility  tem- 
perature curves,  finding  out  what  the  electromotive  force  would 
be  for  the  actual  temperature  in  question. 

1  For  further  details  in  connection  with  the  question  of  cells,  cf.  Nernst, 
Sitzimgsber :  Berlin,  Akad.,  1909. 


PART   III 


CONSIDERATIONS   BASED   UPON   THERMODYNAMICS 
AND   STATISTICAL   MECHANICS 

INTRODUCTORY. 

IN  this  part  a  brief  account  is  given  of  the  behaviour  of 
material  systems  under  the  influence  of  radiation,  which 
behaviour,  when  the  radiation  consists  of  short  waves  (visible 
or  ultra-violet),  is  included  in  the  term  Photochemistry.^  As  a 
matter  of  fact,  however,  recent  advances,  due  notably  to  Trautz 
(Zeitsch.  physik.  Chem.,  64,  66,  and  67;  Zeitsch.  Elektrochem., 
15  and  18)  and  Kriiger  (Zeitsch.  Electrochem.^  18,  1911),  have 
emphasised  the  importance  of  long  wave  radiation  as  the 
possible  origin  of  "  ordinary  "  or  thermal  reactions,  so  that  it 
will  be  evident  that  radiation  in  general  is  of  fundamental 
significance  from  the  chemical  standpoint.  Radiation  chemistry 
has  been  developed  by  the  aid  of  the  ordinary  kinetic  theory 
and  by  thermodynamics,  but  in  addition  to  these  two  modes 
of  investigation  a  third  mode,  by  the  application  of  generalised 
kinetic  theory  or  statistical  mechanics  in  a  new  and  modified 
form,  as  expressed  in  the  quantum  theory  of  Planck,  has 
within  recent  years  proved  of  the  utmost  service  in  physical 
and  chemical  investigation,  notably  in  the  hands  of  Einstein, 
Nernst,  Lindemann,  and  others.  The  first  chapter  of  this  Part 
will  be  devoted  to  the  survey  of  Photochemistry  along  what 
may  be  now  regarded  as  classic  lines.  In  the  second  chapter 
the  concept  of  the  unitary  theory  of  radiant  energy — the 

1  The  volume  on  Photochemistry ',  by  H.  S.  Sheppard,  in  this  series  of 
text-books,  has  recently  appeared.     Further  details  may  be  obtained  there. 


396       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

quantum  theory — and  some  of  its  applications  will  be  dis- 
cussed. It  must  be  remembered  that  this  theory  is  of  very 
recent  growth,  but  already  it  has  established  itself  as  affording 
the  best  explanation  of  the  phenomena  connected  with  such 
apparently  diverse  things  as  the  distribution  of  energy  in  the 
spectrum  of  a  full  radiator  and  the  specific  heat  ot  solids  in 
relation  to  temperature ;  and  further,  it  bids  fair  to  become 
the  most  comprehensive  instrument  yet  known  for  the  explana- 
tion of  physical  and  chemical  phenomena  in  general. 


CHAPTER   I 

Behaviour  of  systems  (in  equilibrium  and  not  in  equilibrium)  exposed  to 
radiation  —  Photochemistry  —  Thermodynamic  treatment  of  photo- 
chemical reactions. 

THE  SOURCE  OF  LIGHT  RADIATIONS. 

ON  the  basis  of  the  electromagnetic  theory  of  light,  the  vibra- 
tions of  small  charged  particles  called  "  radiators  "  or  "  vibra- 
tors," which  may  be  either  atoms  themselves  or  electrons 
present  in  the  molecules  of  all  substances,  produce  electro- 
magnetic waves  of  their  own  period,  i.e.  light  waves  of  a  given 
colour  which  are  radiated  off  into  space.  The  act  of  radiation 
of  such  waves  represents,  of  course,  a  loss  of  energy  on  the 
part  of  the  vibrating  system,  and  the  movement  of  the  vibrators 
will  be  gradually  damped  down  and  cease  unless  energy  is 
communicated  to  them.  There  are  essentially  two  ways  in 
which  this  energy  supply  may  be  kept  up.  First,  the  tempera- 
ture of  the  body  as  a  whole  may  be  kept  high.  This  means 
that  the  kinetic  energy  of  the  molecules  is  great,  and  the 
consequent  energy  interchange  at  collisions  can  go  to  keep 
the  radiators  in  vibration.  Note  that  no  chemical  change  in 
the  structure  of  the  molecules  is  assumed.  This  type  of 
radiation,  which  is  kept  up  by  purely  physical  means,  is  called 
temperature  radiation.  Any  substance  heated  to  a  sufficiently 
high  temperature  must  give  rise  to  such  temperature  radiation. 
The  higher  the  temperature  the  greater  the  collision  frequency 
of  the  molecules,  and  consequently  the  more  rapid  and  more 
energetic  are  the  vibrations  of  the  radiators.  Rapid  vibration 
means  short  wave  length  or  large  frequency,  and  as  the 
temperature  is  raised  evidently  the  radiations  which  corre- 
sponded at  first  to  long  waves,  i.e.  the  infra-red,  may  be  made 


398       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

to  possess  wave  lengths  corresponding  to  the  visible  spectrum 
range,  and  even  the  ultra-violet.  When  the  temperature  is 
raised  so  high  as  to  cause  the  radiators  to  radiate  off  waves 
of  lengths  lying  within  the  limits  of  the  visible  spectrum 
the  body  will  give  out  light.  Particles  capable  of  vibrating 
so  rapidly  as  this  must  necessarily  possess  very  small  mass, 
and  it  is  generally  considered  that  the  radiators  in  such  cases 
are  electrons.  If  all  the  electrons  had  the  same  period,  the 
light  would  be  of  a  certain  colour.  If,  as  is  generally  the  case, 
the  periods  differ  considerably,  the  total  light  emitted  will  be 
white,  and  on  analysing  by  means  of  a  prism  a  continuous 
spectrum  will  be  obtained. 

Besides  this  purely  temperature  radiation  we  have  what  is 
called  luminescence.  In  this,  which  is  generally  present  in 
incandescent  vapours  and  gases  (ordinary  Bunsen  burner),  the 
supply  of  energy  to  the  vibrating  electrons  comes  from  chemical 
change,  i.e.  intramolecular  changes  in  the  molecules  themselves. 
There  are  several  kinds  of  luminescence,  covering,  in  fact,  all 
possible  sources  of  energy  supply  other  than  mere  collision 
effects  due  to  temperature.  It  must  be  noted  that  luminescence 
may  be  exhibited  by  systems  whose  temperature  is  low,  and, 
in  fact,  it  was  observations  on  such  that  first  showed  the 
necessity  of  assuming  that  causes  other  than  high  temperature 
might  be  the  origin  of  the  visible  light.  Thus  Pringsheim  has 
obtained  effects  in  a  photographic  plate  from  a  CS2  flame  whose 
temperature  was  only  150°  C.  Pure  temperature  radiation 
would  in  this  case  (at  150°  C.)  have  produced  no  photographic 
effect,  for  the  temperature  radiation — which  must  always  be 
present  even  with  luminescence — would  correspond  to  feeble 
and  slow  vibrations  far  in  the  infra-red. 

We  shall  return  again  to  the  subject  of  temperature  radia- 
tion and  luminescence  after  considering  some  of  the  evidence 
that  it  is  a  charged  particle  of  dimensions  much  smaller  than 
an  atom,  i.e.  an  electron,  which  gives  out  visible  and  ultra- 
violet rays  by  its  vibration. 

Suppose  we  have  two  electric  charges  of  opposite  sign, 
+  e  and  —  e,  whose  distance  apart  undergoes  a  periodic 
change  of  amplitude  a,  then  according  to  Hertz  (quoted  in 


OPTICAL    VIBRATORS   (RESONATORS]          399 

Drude's  Optics,  p.  530),  the  electromagnetic  energy  emitted 
in  a  half  period  is — 

SA3^2 

where  A  is  the  wave  length  of  the  radiation  emitted.  Hence 
the  amount  of  energy  radiated  per  second  from  such  a 
charge  is — 

L  =  !7T4^—     or    ^A4<r-r— 
A31  A4 

where  T  is  the  periodic  time  of  a  single  complete  vibration, 
or  -  is  the  frequency  of  the  radiation  and  c  is  its  velocity  (i.e. 

the  velocity  of  light,  namely,  3  X  10*°  cms.  per  second),  these 
terms  being  connected  by  the  well-known  relation — 

A 

c  =  -A  =  vX 

where  v  is  the  frequency  of  vibration. 

Now,  for  the  particular  case  of  incandescent  sodium  vapour, 
which  happens  to  be  mainly  a  luminescent  source  and  only 
partly  a  temperature  source,  Wiedemann  has  shown  that  the 
energy  emitted  per  second  in  the  two  D  lines  by  i  gram 
of  sodium  is  3210  gram-calories  or  13*45  X  io10  ergs.  The 
sodium  vapour  may  be  regarded  as  partially  ionised ;  that  is 
to  say,  a  neutral  atom  has  given  off  an  electron  (charge  —  e), 
the  remainder,  the  positive  ion,  having  the  charge  +  *• 
Suppose  that  the  electron  vibrates  with  respect  to  the  positive 
ion,  which  latter  happens  to  be  much  larger  in  mass  and  size, 
and  may  be  regarded  as  at  rest.  Then  from  each  atom  of 
sodium  vapour  thus  ionised  we  would  expect  to  obtain  the 
energy  L  per  second.  From  Perrin's  determination  of  the 
number  of  molecules  in  i  gram-molecule,  viz.  6  X  io23  (in 
round  numbers),  it  is  evident  that  there  are  6  X  io23  atoms 
in  23  grams  of  sodium  vapour,  or  2*6  X  io22  atoms,  in 
i  gram.  Since  each  ionised  atom  when  radiating  gives  rise  to 
a  radiation  energy  L  per  second,  then,  if  all  the  atoms  were 


400       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

thus  radiating,  the  energy  emitted  per  second  from  i  gram 
of  sodium  would  be — 

-6X  io22 


This  is  evidently  a  maximum  value,  since  ionisation  is  probably 
not  complete.  It  may  give  the  correct  order  of  magnitude, 
however.  Equating  this  to  Wiedemann's  experimental  value, 
we  obtain  — 


o22  =  13-45  X 

The  quantity  e,  the  unit  charge,  is  approximately  4'6  X  io~10 
electrostatic  units  (cf.  Vol.  L,  Part  L,  Chap.  I.).  Further,  the 
mean  wave  length  of  the  D  lines  emitted  by  incandescent 
sodium  vapour  is  0-000589  cm.  Hence  for  the  amplitude 
of  the  vibrating  particle,  we  obtain  — 

a  =  8  X  10-1°  cm. 

The  diameter  of  an  atom  is  of  the  order  io~8  cm.  The 
amplitude  of  the  vibrating  particle  is  thus  considerably  smaller 
than  the  diameter  of  an  atom  itself.  It  seems  reasonable 
to  assume,  therefore,  that  the  actual  vibrating  agent  is  an 
electron,1  which  is  known  to  have  the  mass  of  only  ygJoo  °f  a 
sodium  atom. 


RADIATION  DUE  TO  TEMPERATURE  ALONE  (AS  OPPOSED 
TO  LUMINESCENCE). 

There  are  several  important  relationships  or  laws  which 
may  be  mentioned  in  connection  with  temperature  radiation. 
Since,  however,  these  form  a  part  of  physical  optics,  a  brief 
statement  must  here  suffice.  For  their  deduction,  or  theoretical 
significance  and  practical  applicability,  a  textbook  on  physical 
optics  is  to  be  consulted,  e.g.  Wood's  Physical  Optics. 

1  Sec  F.  A.  Lindemann,  Verhl.  d.  Phys.  Gescll.,  13,  482,  1911,  for  an 
alternative  method  of  treatment.  1 


THE   FULL   RADIATOR  4°* 

Laws  of  Balfonr  Stewart  and  Kirchhoff. 

The  first  Stewart-KirchhofF  Law  states  that  light  of  any 
given  wave  length  emitted  by  a  (gaseous)  body  can  also  be 
absorbed  by  that  body  at  a  lower  temperature.  This  law 
at  once  affords  a  reasonable  explanation  of  the  absorption 
lines — the  Frauenhofer  lines — in  the  sun's  spectrum.  The  sun 
itself  is  at  an  intensely  high  temperature,  and  is  surrounded 
by  a  gaseous  atmosphere  of  similar  constitution,  but  at  a 
lower  temperature.  This  atmosphere,  which  contains  several 
metallic  vapours,  such  as  that  of  sodium,  present  in  it, 
absorbs  to  a  certain  extent  the  sodium  light  given  out  by 
the  sun  itself,  and  hence  the  Frauenhofer  dark  lines  corre- 
sponding exactly  to  the  yellow  D  lines  of  incandescent  sodium 
vapour.  The  Frauenhofer  lines,  with  the  aid  of  the  above 
law,  allow  us  to  ascertain  the  chemical  constitution  of  the 
sun.  Of  course,  the  light  emitted  by  an  incandescent  vapour 
is  by  no  means  purely  temperature  radiation.  Luminescence 
is  likewise  present.  The  Stewart-Kirchhoff  Law,  therefore, 
applies  /«  general.  The  Stewart-Kirchhoff  Second  Law  states 
that  the  ratio  of  the  emissive  power  of  the  body  to  its  absorp- 
tive power  is  a  function  of  the  temperature  only,  and  is  the 

same  for  all  bodies  emitting  "  temperature  radiation."     We 

•p 

can  write  this  in  the  form  —  =  constant.     By  the  term  "  emis- 

A 

sive  power  "  is  meant  the  intensity  of  radiation  of  given  wave 
length  emitted  at  a  given  temperature.  The  absorptive  power 
of  a  body  is  the  fraction  of  incident  radiation  absorbed  by  the 
body.  A  perfectly  absorbing  or  "  black  "  body  is  one,  as  the 
name  implies,  for  which  A  is  unity.  If  the  emissive  power 
of  a  black  body  at  a  given  temperature  is  ^,  then,  since  A  =  i, 

E 

e  =  •— .     In  other  words,  the  emissive  power  of  a  black  body 
A 

or  "  full  radiator  "  is  equal  to  the  ratio  of  the  emissive  to  the 
absorptive  power  of  any  body  at  the  same  temperature.  It  is 
to  be  remembered  that  this  is  only  true  of  temperature 
radiation. 

T.P.C. — ii.  2  D 


4o?       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

Stefan's  Law  for  Total  Radiation. 

According  to  Stefan,  the  total  amount  of  energy  radiated 
from  a  body  which  is  giving  out  a  series  of  different  wave 
lengths,  i.e.  an  entire  spectrum,  such  radiation  being  due  to 
temperature  only,  is  proportional  to  the  fourth  power  of  the 
absolute  temperature  of  the  body,  i.e.  oc  T4.  Stefan  was  led 
to  this  conclusion  from  a  consideration  of  Tyndall's  data  upon 
the  radiation  from  a  platinum  wire.  The  quantities  of  energy 
radiated  per  second  at  the  temperatures  1200°  C.  and  525°  C. 
respectively  were  measured  and  the  ratio  found  to  be  117. 

(i  200  -f-  27"2\* 
-1    was  equal  to  ir6.     The 
525  +  273' 

law  has  been  tested  frequently  since,  and  the  very  accurate 
work  of  Lummer  and  Pringsheim  has  shown  its  applicability 
over  a  wide  range  of  temperature.  (For  details  see  Preston's 
Heat,  latest  edition,  and  the  Annalen  der  Physik  for  more 
recent  communications.)  If  we  regard  the  radiation  of  the 
sun  as  a  pure  temperature  effect  (which  is  only  approximately 
true),  bolometic  measurements  give  the  amount  of  energy 
radiated  per  second,  and  hence,  by  applying  Stefan's  Law,  the 
temperature  of  the  sun  may  be  calculated.  The  result  works 
out  as  6200°  absolute;  this,  however,  is  necessarily  approxi- 
mate only.  Stefan's  Law  has  been  deduced  by  Boltzmann  from 
thermodynamical  considerations  applied  to  radiation — a  fact 
of  fundamental  importance  (see  Preston's  Heat  or  Drude's 
Optics,  or,  in  fact,  any  work  on  the  theory  of  radiation). 

Wieris  Displacement  Law. 

In  Stefan's  Law  we  only  take  account  of  the  total  energy 
emitted  by  a  source.  We  now  come  to  the  much  more 
complex  question — complex  even  when  we  restrict  ourselves 
to  temperature  radiation  alone,  and  quite  beyond  our  knowledge 
at  present  in  the  case  of  luminescence  effects — namely,  the 
distribution  of  the  energy  radiated  among  the  various  wave 
lengths  emitted.  Temperature  *  radiation  always  gives  rise  to 
a  complete  spectrum.  The  energy  from  a  source  thus  radiating 


IV TEN'S   DISPLACEMENT  LAW  403 

is  found  to  vary  with  the  wave  length  emitted,  but  does  not 
increase  or  decrease  regularly  as  we  pass  along  the  spectrum. 
Instead,  the  energy-wave-length  curve  passes  through  a  maxi- 
mum at  a  certain  wave-length  region,  the  temperature  of  the 
source  having  a  definite  value  (Fig.  90).  By  altering  the 
temperature  of  the  source,  the  position  of  the  maximum  is 
found  to  alter  correspondingly.  The  direction  of  the  change 
is  such  that  the  maximum  shifts  towards  the  shorter  wave- 
length end  of  the  spectrum  as  the  temperature  rises.  This 
phenomenon  of  shift  of  maximum  is  taken  account  of  in  Wien's 
Displacement  Law.  If  we  denote  by  Amax  the  wave  length  corre- 
sponding to  the  energy  maximum,  and  by  T  the  absolute 
temperature  of  the  radiating  black  body,  Wien's  law  states  that 
Amax  X  T  =  constant.  Some  of  Lummer  and  Pringsheim's 
data  in  support  of  this  are  given  in  a  subsequent  table  (the 
most  recent  and  most  thorough  investigation  up  to  date  (1914) 
upon  the  radiation  from  a  black  body  is  that  of  W.  W.  Coblentz, 
J3nll.  Bur.  Standards •,  Jan.,  1914).  Wien  also  deduced — by 
thermodynamic  means — an  important  relationship  between  the 
maximum  intensity  and  the  absolute  temperature.  If  we 
denote  this  intensity  by  Emax,  Wien's  relation  is — 

^  =  constant 

By  E  is  meant  the  quantity  of  energy  emitted  per  second 
from  a  given  narrow  wave-length  region  containing  A,  the  energy 
being  absorbed  by  a  thermopile  or  bolometer,  and  the  amount 
of  the  energy,  i.e.  the  magnitude  of  E,  measured  by  the  current 
produced  as  indicated  by  a  very  sensitive  galvanometer.  Some 
data  in  connection  with  this  law  are  also  included  in  the  table. 
It  may  be  mentioned  that  a  totally  absorbing  or  perfectly 
black  body  has  been  very  nearly  realised  in  practice  by  using 
a  hollow  blackened  sphere  or  cylinder  of  metal  (heated  elec- 
trically) possessing  a  small  opening  through  which  the  light 
passes  and  is  reflected  many  times  from  side  to  side  until  it 
is  practically  all  absorbed,  for  of  course  each  reflection  is 
accompanied  by  a  considerable  amount  of  absorption.  It 
must  be  remembered,  of  course,  that  though  we  speak  of 


404       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 
radiation  from  a  black  body,  the  word  "  black "  is  not  to  be 


140- 


130- 


Wave-length. 
(From  ShepparcFs  "  Photo-chemtstry.") 

FIG.  90. 

taken  as  referring  to  absence  of  colour.     At  high  temperatures 
metals  become  "  red "   and   finally   "  white "   hot,  though  all 


TEMPERATURE  RADIATION 


405 


the  time  they  may  be  emitting  temperature  radiation  only 
and  functioning  as  "  black  "  bodies.  A  better  term  is  "  full 
radiator."  In  Lummer  and  Pringsheim's  experiments  (Annal. 
d.  Pliysik)  6,  192,  1901)  the  spectrum  was  produced  by  refrac- 
tion through  a  prism  of  fluorspar,  which  is  very  transparent, 
particularly  to  infra  red  rays.  As  lenses  could  not  be  used, 
the  image  of  the  slit  was  formed  by  means  of  a  concave 
mirror.  A  Lummer-Kurlbaum  linear  bolometer  was  employed 
to  measure  the  radiant  energy,  the  width  of  metal  strip 
exposed  being  o'6  mm.  and  the  thickness  o'ooi  mm.  The 
results  were  reduced  to  the  normal  spectrum,  that  is  the 
spectrum  produced  by  a  grating,  the  constants  for  fluorspar 
having  been  previously  determined  by  Paschen  (Annal.  d. 
Physik)  53,  301,  1894).  As  the  strip  of  platinum  in  the  bolo- 
meter possesses  width,  the  energy  measured  will  not  correspond 
to  a  single  wave  length,  but  to  a  small  region  lying  between  A 
and  \-\-d\.  Also  as  the  width  of  the  slit  is  finite,  the 
spectrum  is  not  quite  pure. 

LUMMER  AND  PRINGSHEIM'S  RESULTS. 


^nmx>  *"•£•  the  wave  length 

EUUVX,  *"•£•  maximum 

Tabs. 

corresponding  to 

intensity  in  arbitrary 

^•max  T. 

mfx  x  io17 

maximum  radiation. 

units. 

T 

621'2 

4-S3/* 

2-O26 

2814 

2190 

908-0 

3-28 

13-66 

2980 

2208 

1094*5 

2-71 

34'o 

2966 

2164 

1259-0  |                  2-35 

68-8 

2959               2176 

1460-4  |                  2-04 

J45'° 

2979 

2184 

1646*0 

1-78                          270-6 

2928 

2246 

Mean  of  these  and  other  data 

2940 

2188 

The  energy  max.  of  the  solar  radiation  lies  in  the  yellow- 
green  region,  and  by  making  use  of  the  expression  AmaxT  = 
constant,  Langley  calculated  the  temperature  of  the  sun  to  be 
5880°  abs.,  which  agrees  fairly  well  with  that  calculated  from 
Stefan's  Law.  It  has  been  found  that  the  relation  AaaxT  = 
constant  holds  more  accurately  in  general  (for  substances  such 
as  platinum,  iron  oxide,  copper  oxide,  carbon,  none  of  which 


406       A   SYSTEM   OF  PHYSICAL   CHEMISTRY 

are  of  course  "  perfectly  black  ")  than  the  other  laws  quoted. 
It  is  therefore  of  considerable  practical  applicability. 

LAWS  OF  THE  DISTRIBUTION  OF  ENERGY  THROUGHOUT 
THE  SPECTRUM. 

We  have  already  used  the  phrase  "  energy  corresponding 
to  a  given  wave  length."  This  of  course  really  refers  to  the 
energy  emitted  between  the  wave  lengths  A  and  A  -}-d\.  It 
is  impossible  to  isolate  an  actual  single  vibration  experi- 
mentally. The  method  of  procedure  is  to  isolate  a  very  small 
portion  of  the  spectrum  lying  between  two  wave  lengths  very 
close  together,  and  measure  the  energy.  Several  radiation 
formulae  dealing  with  the  distribution  of  energy  throughout 
the  spectrum  have  been  proposed.  The  most  important  of 
these  are  the  formulas  of  Rayleigh,  of  Wien,  and  of  Planck. 
Rayleigh's  formula  is  as  follows — 

CT     -— 
EA  =  ~F'"    ' 

Wien's  is 

C     ~ 


where  C  and  c'  are  constants.  The  latter  expression  has  been 
carefully  investigated,  notably  by  Lummer,  Pringsheim,  and 
Paschen,  who  have  found  that  it  holds  with  a  great  degree  of 
accuracy  over  a  wide  region,  but  not  over  the  entire  region 
accessible  to  measurement.  With  very  long  waves  and  at  high 
temperatures  the  expression  ceases  to  accurately  reproduce 
the  experimental  values  of  EA ;  Rayleigh's,  on  the  other  hand, 
holds  in  this  region,  but  not  in  the  short  wave  region.  These 
distribution  laws  mentioned  are  therefore  not  the  last  word 
on  the  subject.  Yet  another  distribution  law  has  been  put 
forward  by  Planck,  though  the  ideas  lying  at  the  base  of  this 
differ  to  such  an  extent  from  foregoing  considerations,  that  a 
discussion  of  this  expression  must  be  postponed  until  we  come 
to  study  Planck's  revolutionary  concepts  regarding  radiant 
energy,  the  so-called  unitary  theory  of  energy. 


RADIATION  FORMULAE  4°7 

A  point  still  remains  to  be  mentioned  in  connection  with 
temperature  radiation.  It  must  be  remembered  that  the 
above  expressions  (Rayleigh's,  Planck's,  and  Wien's)  hold  only 
for  bodies  emitting  continuous  spectra,  i.e.  all  wave  lengths. 
For  line  spectra,  such  as  one  finds  in  the  case  of  incandescent 
ases  and  vapours,  no  relation  has  yet  been  found  between  A 
and  E.  Pfliiger  has  shown  by  bolometric  measurements  in  the 
case  of  line  spectra,  that  the  largest  deflections  were  obtained 
in  the  ultra-violet,  i.e.  the  ultra-violet  lines  are  "  hotter  "  than 
the  infra-red.  This  is  in  direct  opposition  to  the  energy 
distribution  in  a  continuous  spectrum  (due  to  temperature 
radiation). 

NOTE.  —  The  various  laws  which  have  here  been  very 
briefly  discussed,  are  much  more  fully  dealt  with  in  Preston's 
Heat  (Cotter's  edition)  and  also  by  Wien,  Lummer,  Pringsheim, 
and  Rubens,  in  vol.  2  of  the  Reports  of  tJie  International 
Congress  of  Physics,  Paris,  1900. 

LUMINESCENCE. 

This  term  is  given  to  radiation  of  light  waves,  the  vibrating 
electrons  being  kept  in  motion  by  processes  dependent  on 
factors  other  than  temperature  alone.1  There  are  various 
dnds  of  luminescence,  as  the  table  on  p.  408  will  make  clear. 

These  various  types  of  luminescence  may  be  very  briefly 
described. 

By  fluorescence  is  meant  the  phenomenon  of  giving  out 
ight  when  under  the  influence  of  a  beam,  this  emitted  light 
differing  in  wave  length  from  that  causing  the  fluorescence. 
Vhen  the  exposure  to  the  beam  ceases,  fluorescence  also 
ceases,  and  differs  in  this  from  phosphorescence,  which  is  the 
continued  emission  of  light  after  the  external  radiation  is  cut 
off.  There  are  many  fluorescent  substances,  e.g.  calcium 
fluoride,  uranium  glass,  liquids  like  petroleum,  solutions  of 
eosin,  quinine  sulphate,  and  many  organic  dyestuffs,  such  as 

1  If  the  total  radiation  from  a  body  is  greater  than  that  given  by 
Stefan's  Law,  then  the  body  must  be  emitting  light  by  luminescence  as 
well  as  the  necessary  temperature  radiation. 


4o8       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 


litmus,  in  general.  Wood  has  also  found  that  the  vapours  of 
iodine,  sodium,  and  mercury  fluoresce.  A  law  due  to  Stokes, 
which  was  formerly  thought  to  be  absolutely  general,  but  with 
regard  to  which  several  exceptions  have  been  found,  states 
that  in  fluorescence  the  wave  length  of  the  emitted  light  is 
always  longer  than  the  incident  light.  Thus  ultra-violet  light 
can  be  absorbed  by  several  of  the  substances  above-named, 
and  given  out  as  blue-green  fluorescence.  Important  excep- 
tions are  Magdala  red  dye,  chlorophyll,  eosin  and  sodium 
vapour  (the  latter  giving  out  a  yellow-green  fluorescence  when 
stimulated  by  yellow  light). 


Source  or  Origin  of  the  Radiation. 


1.  Radiation    caused   by  ex- 
posure   of   the    body    to 
external  radiation. 

2.  Gentle   heating   (not  to   a 

sufficiently  high  tempera- 
ture to  give  temperature 
radiations  of  short  enough 
wave  length  to  be  visible 
(the  latter  requires  at  least 
360*  C.)). 

3.  Friction  or  crystallisation. 

4.  Chemical  reactions. 

5.  Electrical    charge   or   dis- 

charge. 


Name  given  to  the  Radiation. 


Photoluminescence^     which     is 
divided  into : — 

a.  Fluorescence. 

b.  Phosphorescence. 


Thcrmolnminescence. 


Triboluminescmcc. 

Chemiluminescencc. 

Electroluminescence. 


Phosphorescence  is  shown  by  phosphorus,  the  sulphides 
of  the  alkaline  earths,  diamond,  and  several  varieties  of 
calcium  fluoride.  It  is  also  exhibited  by  decomposing  organic 
matter,  though  this  perhaps  should  come  under  the  heading 
of  chemiluminescence.  The  Stokes .  Law  for  fluorescence 
also  holds  for  phosphorescence.  A  very  important  point  has 
been  brought  out  in  connection  with  the  phenomenon  of 


LUMINESCENCE  409 

phosphorescence,  namely,  that  impure  substances  are  much 
more  active  than  pure.  Thus  an  addition  of  0*00008  parts  of 
CuO  to  i  part  CaO  gives  a  very  bright  phosphorescence.  In 
general  we  might  say  that  mixed  crystals,  i.e.  solid  solutions, 
are  good  phosphorescent  or  fluorescent  substances.  In  fact,  the 
property  of  fluorescing  in  a  vacuum  tube  while  the  discharge 
is  passing,  is  taken  to  be  evidence  that  the  substance  in 
question  is  a  mixed  crystal.  We  are  indebted  chiefly  to 
Lenard  and  his  pupils,  as  well  as  to  Urbain,  for  our  knowledge 
of  this  phenomenon. 

As  regards  thermoluminescence  there  is  little  to  be  said. 
Certain  substances  on  being  gently  warmed  give  out  light 
waves.  This,  however,  only  happens  if  the  substance  has 
previously  been  exposed  to  light.  The  sulphides  of  the 
alkaline  earths  exhibit  this  phenomenon,  which  is  evidently 
closely  related  to  the  property  of  phosphorescence. 

Triboluminescence  and  crystalloluminescence  are  exhibited 
by  many  ordinary  substances.  Sugar  crystals,  for  example, 
when  crushed  in  the  dark,  emit  light.  The  same  is  the  case 
with  crystals  of  uranium  nitrate.  The  process  of  solidifica- 
tion of  a  melted  substance  is  also  in  some  cases  (e.g.  fused 
silver)  accompanied  by  light  emission. 

Chemiluminescence  is  a  phenomenon  which  is  fairly  general 
in  chemical  processes.  Trautz  (Zeitsch.  Elektrochem.^  14,  453, 
.1908)  gives  several  instances.  The  precipitation  of  sodium 
chloride  from  solution  by  means  of  hydrochloric  acid ;  the 
dissolution  of  solid  sodium  hydrate  in  hydrochloric  acid ;  the 
reactions  involving  the  evolution  or  absorption  of  gases,  chiefly 
oxygen,  as  in  the  case  of  pyrogallol  shaken  with  air,  and 
formaldehyde  with  hydrogen-peroxide,  as  well  as  the  complex 
oxidations  which  go  on  during  putrefaction.  Luminescence 
also  occurs  in  the  interaction  of  oxygen  with  phosphorus 
trioxide,  with  alkalies  and  alkaline  earths,  as  well  as  by  the 
interaction  of  halogens  with  these  metals.  Besides  these 
heterogeneous  reactions,  light  effects  have  been  observed  in 
homogeneous  gaseous  systems.  Thus  when  a  rapid  stream  of 
acetylene  mixed  with  bromine  vapour  passes  into  a  cylinder,  a 
feeble  green  flame  can  be  observed,  the  temperature  of  which 


410       A   SYSTEM  OF  PHYSICAL    CHEMISTRY 

is  very  low.  "Cold"  flames  in  general  are  instances  of 
chemiluminescence.  Trautz  sums  up  the  characteristics  of 
this  phenomenon  as  follows :  Chemiluminescence  is  very 
general.  Its  intensity  increases  (cet.  par.)  (i)  with  the  heat 
effect  of  the  reaction  involved,  (2)  with  the  velocity  of  the 
reaction,  and  (3)  enormously  with  rise  of  temperature.  The 
colour  of  the  luminescence  is  dependent  on  the  reacting 
system,  but  is  independent  of  the  reaction  velocity  and  of  the 
temperature. 

Electroluminescence  is  a  familiar  phenomenon  since  the 
introduction  of  the  Geissler  tubes  for  making  gases  incan- 
descent for  spectroscopic  purposes.  It  has  been  observed 
that  the  luminescence  in  some  cases  continues  for  a  short  time 
after  the  discharge  has  ceased,  as  in  the  case  of  phosphor- 
escence. The  luminescence  emitted  by  bodies  (generally 
solids)  when  placed  in  a  vacuum  tube  under  the  action  of 
cathode  rays,  to  which  reference  has  already  been  made  under 
fluorescence,  may  also  be  regarded  as  electroluminescence 
effects,  as  may  also  be  the  effects  on  a  zinc  sulphide  screen 
when  bombarded  by  X-rays,  or  radioactive  a  and  j8  particles. 

We  have  now  discussed  the  problem  of  the  different 
methods  of  light  production.  Under  the  next  heading  we 
shall  deal  with  photochemistry  proper,  i.e.  the  study  of 
chemical  effects  brought  about  in  systems  exposed  to  radiation. 


PHOTOCHEMISTRY. 

General  (cf.  R.  Luther  (Zeitsch.  ElektroeJiem.,  14,  445, 1908)). 
— Reactions  which  are  either  initiated  or  accelerated  by 
radiant  energy,  the  wave  length  of  which  corresponds  either 
to  the  visible  spectrum  or  to  the  ultra-violet,  are  termed  photo- 
chemical reactions.  Such  reactions  are  probably  much  more 
general  than  is  usually  believed.  Perhaps  the  most  familiar 
instances  are  the  effect  of  light  on  silver  salts,  a  reaction  which 
is  the  basis  of  photography,  and  the  effect  of  light  on  the 
chlorophyll  of  a  plant  leaf,  which  enables  the  plant  to  absorb 
oxygen,  carbon  dioxide,  and  water,  and  use  these  to  build 
up  the  complex  organic  substances  which  are  found  in  plants. 


BEER'S   LAW  411 

It  is  rather  an  arbitrary  distinction,  of  course,  to  limit  photo- 
chemical reactions  to  the  waves  of  the  visible  spectrum  and 
the  ultra-violet  region,  for  infra-red  waves  also  represent  radiant 
energy,  though  their  influence  is  usually  regarded  as  belonging 
to  heat  effects.  The  wave-length  limits  for  photochemical 
changes  are  therefore  SOO/A/A  (red)  to  circa  300  or  less  (ultra- 
violet). As  a  rule,  the  shorter  wave  lengths,  i.e.  the  ultra-violet 
region,  are  much  more  chemically  active  than  the  visible  region  . 
The  generalisation  which  is  at  the  base  of  photochemistry  is, 
that  only  those  waves  which  are  absorbed  by  the  substance  can 
be  chemically  active.  This  was  first  stated  by  Theodor  von 
Grotthus  more  than  a  century  ago.  That  no  simple  connection 
exists  between  absorption  and  chemical  action  is  at  once 
shown  by  the  fact  that  many  cases  are  known  where  a  strong 
light  absorption  corresponds  to  practically  no  detectable 
chemical  reaction,  and  on  the  other  hand  a  marked  chemical 
action  takes  place  in  cases  where  absorption  is  apparently 
slight.  The  expression  known  as  Beer's  Law  (1852)  of  light 
absorption  depends  essentially  on  the  validity  of  the  assump- 
tion that  absorption  is  due  to  the  number  of  individuals  in  a 
given  layer,  each  acting  per  se.  Thus  if  a  solution  is  diluted 
to  twice  its  volume  and  2  cms.  of  this  latter  solution  absorb 
to  the  same  extent  as  i  cm.  of  the  original  solution,  the 
substance  obeys  Beer's  Law.  This  principle  may  be  expressed 
in  the  form  — 


where  I0  is  the  initial  intensity  of  the  light,  I  is  the  intensity 
after  passage  through  a  layer  of  d  cms.,  the  concentration  of 
the  absorbing  substance  being  c,  and  k  the  absorption  constant 
(coefficient).  If  we  take  two  substances  at  two  concentrations 
c±  and  c2,  and  if  d±  and  d%  are  so  chosen  that  the  ratio  I  to  I0 
is  the  same  in  both,  then  — 


Beer's  Law,  however,  does  not  hold  in  all  cases. 

As  exposure  proceeds,  it  is  found  in  general  that  the 
chemical  actions  produced  thereby  also  proceed.  Photographic 
plates  become  darker  the  longer  the  exposure.  A  striking 


4t2       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

phenomenon  known  as  solarisation,  however,  has  been  observed, 
namely,  that  after  long  exposure  to  bright  sunlight  the  image 
of  the  sun  appears  on  the  negative,  light  on  a  dark  ground. 
Other  instances  of  the  reversal  effect  are  known,  but  no  satis- 
factory explanation  has  yet  been  offered.  For  details,  cf. 
Sheppard  and  Mees'  Investigations  on  the  Theory  of  the  Photo- 
graphic Process  •,  Longmans,  Green  &  Co.,  1907.  Turning  to 
the  question  of  reaction  velocity  an  important  point  arises, 
namely,  what  is  the  connection  between  velocity  and  the 
intensity  of  the  light  ?  It  has  been  found,  ceteris  paribus,  the 
reaction  velocity  is  directly  proportional  to  the  light  intensity. 
This,  however,  can  scarcely  be  regarded  as  an  absolutely 
accurate  statement,  for  cases  are  known  in  which  the  reaction 
velocity  increases  more  slowly  than  the  light  intensity  when 
the  latter  is  increased.  At  this  stage  we  have  to  distinguish 
two  light  effects,  i.e.  two  different  ways  in  which  the  light  may 
act.  First,  it  may  simply  act  as  a  catalysing  agent.  That  is 
to  say,  it  may  simply  accelerate  a  reaction  which  would  of 
itself  proceed  slowly  in  the  dark.  It  can  also  act  as  a  negative 
catalyst.  Secondly,  the  light  may  actually  originate  a  reaction 
which  would  otherwise  not  go  at  all,  or  it  may  alter  the  course 
of  a  reaction,  different  end  products  being  obtained  according 
as  to  whether  a  given  reaction  is  allowed  to  take  place  in  the 
light  or  in  the  dark.  Of  course,  optical  catalytic  effects  may 
be  superimposed  on  the  latter  case  at  the  same  time. 

As  regards  the  catalytic  effect,  we  can  formulate  the  process 
as  follows  :  — 

Suppose  two  substances  A  and  B  react  with  each  other, 
then,  according  to  the  Law  of  Mass  Action,  the  rate  at  which 
this  reaction  goes  on  in  the  dark  is  — 

«  [AJ-fB]' 

where  a  and  b  determine  the  order  of  the  reaction. 

If  now  the  same  reaction  takes  place  in  the  light,  we  may 
write  for  a  given  constant  light  intensity  the  rate  of  the  reaction 
as  — 

cc 


where  a  and  j8  determine  the  order  of  the  reaction  in  the  light. 


PHOTO-VELOCITY  413 

It  is  important  to  note  that  a  and  l>t  etc.,  are  as  a  rule  not  the 
same  as  a  and  j3,  etc. ;  in  other  words,  the  order  of  a  given 
reaction  may  not  be  the  same  in  the  presence  of  light  as  it  is 
in  the  dark.  It  has  been  found,  as  a  matter  of  fact,  that  the 
photochemical  exponents  a  and  j8,  etc.,  are  never  greater  than, 
are  rarely  equal  to,  and  are  usually  less  than  the  corresponding 
a,  b,  etc.,  terms.  Thus  in  light  hydriodic  acid  splits  into 
hydrogen  and  iodine,  viz.  HI  ->  H  -j-  Ii  *•*•  a  monomole- 
cular  reaction,  while  in  the  dark  the  following  takes  place : 
2 HI  — >  H2  +  !£,  i.e.  a  bimolecular  action.  The  photo  reaction 
is  here  a  catalysed  one.  Under  certain  circumstances  it  has 
even  been  found  that  the  photochemical  exponent  comes  out 
/ero,  showing  that  the  rate  is  independent  of  the  concentration 
of  the  substance  taking  part.  This,  however,  only  holds  for 
a  certain  concentration  range.  When  the  dark  and  light 
reactions  are  of  the  same  nature  (the  case  discussed  above), 
and  when  the  rate  of  the  dark  reaction  is  not  negligible,  com- 
pared to  that  in  the  light,  it  has  been  found  that  the  total 
velocity  may  be  written  as  a  simple  sum  of  both,  i.e. — 

Total  velocity  =  ^i[A]«[B]&  +  £2[A]aP? 
If  the  intensity  of  the  light  is  I,  then  in  general- 
Total  velocity  =  *i[A]«[B]6  +  V[A]«[B]^ 

Photochemical  reactions,  as  a  rule,  have  very  small  tem- 
perature coefficients.  So  far  we  have  regarded  the  reaction 
as  going  to  an  end.  Photochemical  effects  have  also  been 
observed  in  reactions  in  which  an  equilibrium  exists  and  is 
measurable.  This  brings  us  to  the  phenomenon  of  photo- 
chemical equilibrium,  or,  as  we  shall  call  it  more  accurately 
later,  "photo-stationary  state"  In  such  cases  the  light  energy 
may  oppose  the  chemical  forces  doing  work  against  them,  and 
thereby  giving  rise  to  a  stationary  state,  which  differs  from  the 
equilibrium  point  in  the  dark.  It  is  no  longer  simply  a  catalytic 
effect.  The  photochemical  stationary  state,  however,  differs 
fundamentally  from  the  ordinary  chemical,  as  one  would  expect, 
in  that,  while  the  latter  represents  a  permanent  state  for  all 
time,  the  photochemical  equilibrium  holds  good  only  as  long 
as  the  light  energy  remains  constant.  On  withdrawal  of  the 


414       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

light  the  reaction,  irreversible,  passes  into  the  ordinary  chemical 
equilibrium  state.  A  notable  instance  of  this  is  to  be  found 
in  the  transformation  of  dianthracene  into  anthracene,  which 
has  been  investigated  by  Luther  and  Weigert,  and  which  we 
shall  discuss  later. 

A  further  peculiarity  which  has  been  noticed  in  connection 
with  photochemistry  is  the  so-called  "  period  of  induction," 
i.e.  an  initial  period  during  which  no  reaction  appears  to  take 
place,  but  at  the  conclusion  of  which  the  reaction  progresses 
in  a  normal  manner.  This  was  first  observed  by  Bunsen  and 
Roscoe  in  the  union  of  hydrogen  and  chlorine.  It  is  really 
due  to  secondary  effects  either  of  a  physical  or  chemical  nature, 
and  will  be  discussed  later.  We  may  now  pass  on  to  consider 
some  of  the  more  important  photo-phenomena  in  detail. 


DIVISION  OF  PHOTOCHEMICAL  REACTIONS  INTO  GROUPS. 

We  shall  here  follow  in  the  main  the  division  of  the  subject- 
put  forward  by  F.  Weigert  (Ahrens  Sammlung,  Bd.  17,  1911), 
and  based  on  thermodynamical  grounds.  According  to  a  well- 
known  principle  of  thermodynamics,  a  reaction  which  takes 
place  in  a  vessel  shielded  from  all  external  influence  proceeds 
in  such  a  direction  that  the  free  energy  of  the  system  decreases, 
and  the  reaction  will  cease  when  the  free  energy  is  a  minimum. 
This  is  the  natural  course  of  the  reaction  under  purely  chemical 
forces.  Such  a  reaction,  in  which  free  energy  is  lost,  may  be 
called  a  reaction  involving  a  free  energy  loss ;  a  decrease  in 
free  energy  means  that  work  has  been  or  may  be  done.  Now, 
when  light  falls  upon  a  system  it  may  act  in  two  ways.  It 
may  cause  the  system  to  carry  out  an  ordinary  reaction,  involv- 
ing a  decrease  in  free  energy,  i.e.  the  reaction  may  go  in  the 
"  natural "  way,  or,  on  the  other  hand,  the  light  may  actually 
oppose  the  chemical  forces  and  cause  a  natural  reaction  to  be 
reversed,  or  may  initiate  a  reaction  opposed  to  the  natural  one. 
In  such  a  case  the  free  energy  of  the  system,  instead  of 
decreasing,  actually  increases.  This  may  be  called  a  free 
energy  conservation  reaction,  or  a  reaction  involving  an 
increase  in  free  energy.  This  sort  of  reaction  will  proceed 


TYPES    OF  PHOTO-REACTIONS  415 

until  the  light  energy  is  no  longer  able  to  overcome  the 
opposing  chemical  forces  which  have  increased  owing  to  the 
increase  in  free  energy,  and  a  stationary  state  of  the  system 
ensues.  This  stationary  state  will  depend  on  the  intensity  of 
the  light,  and  only  secondarily  upon  mass  action,  so  that  such 
a  state  is  to  be  clearly  distinguished  from  an  ordinary  "  chemical 
equilibrium  "  which  is  controlled  entirely  by  the  principle  of 
mass  action.  On  removal  of  the  light  the  reaction  may  reverse 
itself,  and  the  system  return  to  its  original  state.  In  the  case 
of  reactions  involving  gain  of  free  energy,  we  have  therefore 
the  possibility  of  "  reversibility."  We  will  now  restate  the 
above  division  of  photochemical  reactions,  and  will  then 
proceed  to  discuss  some  examples.  Photochemical  reactions 
may  be  divided  into  the  following  classes  : — 

I.  Photo-reactions  involving  a  decrease  in  free  energy. 

In  these  reactions  the  free  energy  at  the  end  is  less  than  at 
the  beginning.     This  group  may  be  further  subdivided  into — 

(a)  Reactions  consisting  of  several  consecutive  reactions, 
the  first  of  which  is  photosensitive,  and  the  others  depend  on 
it  in  a  purely  chemical  manner. 

(b)  Reactions   in  which   the   light   acts  as   a  catalyst  or 
produces  an  actual  catalyst  which  hastens  the  purely  chemical 
photoinsensitive  reaction  between  the  substances  in  the  system. 
This  is  an  instance  of  photocatalytic  reactions. 

II.  Photo-reactions  in  which  the  free  energy  of  the  system  is 
increased  by  the  light. 

In  these  reactions  the  free  energy  at  the  stationary  state  is 
greater  than  at  the  beginning.  This  group  consists  of  trite 
reversible  reactions.  In  these  the  photochemical  reaction  is 
simple  (there  are  no  consecutive  reactions).  When  the  light 
is  withdrawn  the  "  dark  reaction  "  (or  reaction  taking  place  in 
the  dark)  simply  retraverses  the  "  light  reaction  "  (or  reaction 
under  the  influence  of  light).  The  classical  example  of  a 
reaction  of  this  type  is  the  polymerisation  of  anthracene  to 
dianthracene  in  the  light  and  its  depolymerisation  in  the  dark. 
This  type  of  reaction  is  to  be  clearly  distinguished  from 
apparently  reversible  reactions.  In  these  the  reaction  is  complex, 
i.e.  a  photosensitive  reaction  and  therewith  one  or  more  purely 


416       A   SYSTEM  OF  PHYSICAL    CHEMISTRY 

chemical  (photoinscnsitivc)  reactions.  The  return  path  in  the 
dark  differs  from  that  followed  by  the  system  in  the  light,  and 
the  reaction  is  really  irreversible  and  the  free  energy  decreases, 
i.e.  Class  I. 

CLASS  I. — REACTIONS  INVOLVING  A  FREE  ENERGY  Loss. 

Group  (a). — To  this  group  belong  the  interesting  examples 
of  oxidation  and  reduction  of  organic  compounds  investigated 
by  Ciamician  and  Silber.  Under  intense  radiation  it  has  been 
found  for  example  that  nitrobenzene  in  the  presence  of  alcohol 
is  reduced  first  to  phenylhydroxylamine  (which  changes  partly 
into  p.  amido-phenol)  and  finally  to  aniline.  Similarly  the 
nitrobenzene  in  the  presence  of  benzaldehyde  passes  at  first 
into  nitrosobenzene,  the  benzaldehyde  being  simultaneously 
oxidised  to  benzoic  acid. 

An  important  class  of  substances  belonging  to  this  group 
are  the  photochemical  sensitisers  used  in  photography.  The  true 
photo-reaction  is  the  reduction  of  the  silver  bromide  or 
chloride  to  some  subsalt,  a  quantity  of  halogen  being  set  free. 
This  reaction  by  itself,  as  a  matter  of  fact,  as  Luther  has 
shown,  is  a  simple  reversible  process  involving  a  gain  of  free 
energy,  i.e.  in  the  dark  the  halogen  unites  with  the  subsalt  to 
give  the  original  salt,  this  latter  reaction  being  in  the  "  natural " 
direction.  When,  however,  the  silver  salt  is  imbedded  in 
gelatine,  as  in  a  photographic  plate,  the  colour  change  is 
observed  to  take  place  much  more  rapidly,  and  is  now  no 
longer  reversible.  The  gelatine  has  reacted  with  the  liberated 
halogen  according  to  the  equation — 

2AgBr  +  gelatine  +  light  ->  Ag2Br  +  brominated  gelatine 

The  gelatine  is  here  the  photochemical  sensitiser.  By  the 
continuous  removal  of  liberated  bromine  it  has  caused  the 
silver  salt  to  continue  decomposing,  and  at  the  same  time  has 
caused  the  reaction  to  become  irreversible.  The  same  sort  of 
chemical  sensitisation  effect  comes  in  if  some  silver  chloride 
is  placed  in  benzene  and  exposed  to  light.  The  silver  salt 
darkens,  the  benzene  becoming  chlorinated.  On  the  other 
hand,  if  some  silver  chloride  be  placed  in  carbon  tetrachloride 


PHOTO-CATALYTIC  PROCESSES  417 

and  then  exposed  to  light,  the  darkening  is  extremely  slow,  as 
the  liberated  halogen  cannot  react  with  the  carbon  tetrachloride, 
which  is  already  saturated  with  chlorine.  These  instances  of 
chemical  sensitisation  are  to  be  distinguished  from  physical 
sensitisation,  to  be  referred  to  later. 

Group  (b). — Photo-catalytic  reactions. 

In  these  reactions  the  light  simply  hastens  a  process  which 
would  take  place  naturally.  This  is  the  result  of  observation, 
as  far  as  it  goes,  but  it  does  not  tell  us  anything  about  the 
mechanism  of  the  hastening  process.  In  an  attempt  to  advance 
a  little  further  in  this  direction,  Weigert  (Annal.  d.  Physik.^  24, 
246,  1907)  has  put  forward  the  following  suggestions.  The 
accelerating  effect  brought  about  by  the  absorbed  radiation  is 
due  in  the  first  place  to  the  formation  of  a  substance,  this 
primary  reaction  being  a  process  involving  a  gain  of  free 
energy,  and  the  substance  so  formed  catalyses  the  main 
reaction,  which  thus  takes  place  in  a  purely  chemical  manner, 
i.e.  in  the  natural  direction  corresponding  to  decrease  of  free 
energy.  According  to  this  view  the  main  reaction  is  really 
not  photosensitive  at  all,  the  true  photo-reaction  being  the 
formation  of  the  catalyst.  Weigert  suggests  reasons  for  believ- 
ing that,  in  a  gaseous  system  at  any  rate,  the  catalyst  is 
present  as  a  heterogeneous  aggregation  much  larger  than 
molecular  size,  and  the  effects  produced  by  these  aggregates 
belong  therefore  to  heterogeneous  catalysis. 

The  classic  photo-catalytic  reaction  is  that  of  the  union  of 
hydrogen  and  chlorine  to  form  hydrochloric  acid,  according  to 
the  reaction — 

H8  +  C12-»2HC1 

This  reaction  was  first  thoroughly  investigated  by  Bunsen  and 
Roscoe  in  1855  (Ostwald's  Klassiker,  34  and  38).  It  had,  of 
course,  been  the  subject  of  earlier  work,  the  most  important 
conclusion  of  such  work  being  the  statement  known  as  the 
Grotth us-Draper  Absorption  Law,  viz.  only  those  rays  are 
effective  which  are  absorbed.  Draper  had  also  made  the 
extremely  interesting  observation  that  chlorine  which  had 
been  exposed  to  light  had  somewhat  different  properties  from 
T.P.C. — n.  2  E 


418       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

unexposed  chlorine^  in  that  the  previously  exposed  chlorine 
reacted  much  more  rapidly  with  hydrogen  (under  the  influence 
of  light)  than  did  the  unexposed  chlorine  under  the  same 
conditions.  Bunsen  and  Roscoe's  first  step  consisted  in  the 
setting  up  of  an  accurate  actinometer  to  measure  the  extent 
of  the  recombination  of  the  gases  under  various  conditions. 
This  actinometer  is  shown  in  the  figure  (Fig.  91). 

This  gas  mixture  was  confined  over  water  (saturated  with 
the  hydrogen  and  chlorine),  the  water  being  screened  from 
the  light.  When  any  hydrochloric  acid  was  formed  it  dis- 
solved in  the  water,  and  the  meniscus  in  the  horizontal 
capillary  moved  along  a  certain  amount.  The  presence  of 
water  in  the  form  of  vapour  is  a  factor  of  great  importance  in 
connection  with  the  reaction  discussed,  for  if  the  gases  are 
very  dry,  then  even  in  the  presence  of  light  combination  only 


FIG.  91. 

takes  place  slowly.  The  water  molecules  may  therefore  take 
part  in  the  formation  of  the  photochemical  catalyst.  Bunsen 
and  Roscoe  were  the  first  investigators  to  fully  study  the 
phenomenon  called  the  induction  period,  to  which  reference 
has  already  been  made.  In  this  connection  it  was  observed 
that  the  sensitivity  of  the  gaseous  mixture  increased  slowly 
with  time  and  reached  a  maximum  in  most  cases  after  three 
to  six  days.  This  sensitivity  or  induction  maximum  decreased 
enormously  if  small  quantities  of  oxygen  were  added,  and  at 
the  same  time  the  induction  period  is  shortened  by  the  oxygen 
so  that  with  i  "3  per  cent,  oxygen  present  in  the  mixture  no 
period  of  induction  is  shown,  but  the  reaction  takes  place 
steadily  at  a  very  slow  rate.  Also  if  the  chlorine  were 
previously  exposed  to  light,  Mellor  has  shown  that  the 
induction  period  may  be  very  much  shortened ;  the  induction 
maximum  or  sensitivity  of  the  gaseous  mixture  is,  however, 
independent  of  the  previous  illumination.  These  observations 


PHOTO-CATALYTIC  PROCESSES  419 

suggest  that  chlorine  by  the  absorption  of  chemically  active 
rays  is  (partially)  changed  into  an  "  active  state  " ;  but  this 
goes  only  a  very  little  way  towards  explaining  the  very 
complicated  phenomena  of  induction  period  and  induction 
maximum.  The  recent  work  of  Burgess  and  Chapman  (Jour. 
Chem.  Soc,,  89,  1402,  1906)  points  to  the  induction  effects  as 
being  accidental  and  not  really  a  characteristic  of  the  photo- 
chemical process.  They  have  traced  such  effects  to  the 
presence  of  impurities,  especially  ammonia,  in  the  water  present 
or  on  the  walls  of  the  vessel.  Thus  by  using  a  quartz  vessel 
whose  surface  adsorbs  gases  and  water  vapour  to  only  a  slight 
extent  compared  with  glass,  and  by  employing  carefully  boiled 
out  water,  the  induction  period  was  made  to  entirely  disappear. 
The  phenomenon  of  "  photochemical  extinction,"  which  was 
also  discovered  by  Bunsen  and  Roscoe,  is,  according  to 
Weigert,  of  rather  doubtful  significance,1  and  may  be  no  more 
characteristic  of  the  actual  photochemical  process  than  is  the 

i  induction  effect.  For  a  further  discussion  of  such  complica- 
tions which  belong  to  isolated  instances,  the  reader  is  referred 
to  Sheppard's  Photochemistry  (this  series  of  text-books).  As 
an  instance  of  an  inorganic  photo-catalytic  reaction  in  solution, 
one  may  mention  the  oxidation  of  iodine  ion  in  acid  solution 
to  the  uncharged  atomic  or  molecular  state  (Plotnikow,  Zeitsch. 

•  physik.  Chem.,  58,  214,  1907).  This  reaction  goes  of  itself 
slowly  in  the  dark.  The  light  effect  is  therefore  purely  a 
catalytic  one.  The  velocity  is  very  dependent  on  the  presence 
of  other  substances  in  the  system,  such  as  copper  sulphate, 
quinine  and  acridine  salts,  chloroform,  ether,  and  the  nature 
of  the  acid  used.  Plotnikow  also  made  the  interesting 
observation  that  the  blue  and  violet  which  are  in  this  case  the 
chemically  active  rays  are  only  very  slightly  absorbed.  Only 
a  very  small  fraction  of  the  radiant  energy  passing  through  the 
system  is  therefore  used  to  form  the  catalyst.  The  reaction 
must  therefore  be  a  very  sensitive  one.  It  shows,  as  has 
already  been  pointed  out,  how  impossible  it  is  (at  least  at  the 

1  The  recent  work  of  Bodenstein  (Zeitsch.  physik.  Chem.,  85,  297 
(1913),  appears  to  lead  to  quite  new  conclusions  regarding  the  reaction 
just  considered. 


420       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

present  time)  to  connect  chemical  activity  with  magnitude  of 
absorption. 

Another  instance  of  photo-catalysis  in  solution  is  the 
decomposition  of  aqueous  sodium  hypochlorite,  according  to 
the  equation — 

NaOCl->NaCl  +  O 

which  proceeds  according  to  the  monomolecular  law.  One 
other  instance  of  photo-reactions  in  solution  may  be  mentioned, 
namely,  the  oxidation  of  oxalic  acid  in  the  presence  of  iron 
oxalate.  The  ferrous  oxalate  (which  is  red)  oxidises  itself  to 
green  ferric  oxalate,  according  to  the  equation — 

2Fe(C204)  +  H2C204  +  O  ->  Fe2(C2O4)3  +  H2O 

In  sufficiently  intense  light  this  becomes  red  once  more,  the 
action  being  represented  by — 

Fe2(C204)3  +  light  ->  2Fe(C2O4)  +  2CO2 

The  red  ferrous  oxalate  is  thus  reformed,  and  the  total  reaction 
may  be  represented  thus — 

H2C2O4  +  O  ->  H2O  +  2CO2 

The  light  in  this  case  acts  as  a  reducing  agent  on  the  ferri 
salt,  the  oxalic  acid  being  at  the  same  time  oxidised. 

We  have  already  had  occasion  to  refer  to  the  phenomenon 
of  chemical  sensitisation.  Reference  must  also  be  made  to  a 
phenomenon  known  as  optical  sensitisation.  This  consists  in 
making  a  system  sensitive  to  a  certain  range  of  wave  lengths 
by  the  addition  of  some  foreign  substance  (which  does  not, 
however,  absorb  chemically  any  of  the  products)  to  the  system. 
Thus  silver  bromide  on  a  plate  is  sensitive  to  the  short  wave 
length  region  (violet  and  ultra-violet).  It  is  only  very  slowly 
acted  upon  by  the  red.  If,  however,  the  silver  salt  is  impreg- 
nated with  some  dyestuff,  such  as  Congo-red  or  eosin,  the 
plate  is  now  sensitive  to  this  colour.  The  optical  sensitiser 
must  absorb  the  wave  length  to  which  it  is  desired  to  make 
the  silver  salt  sensitive.  Many  dyestuffs  can  be  used  for  this 
purpose  and  they  are  characterised  by  the  fact  that  they 


OPTICAL  SENSITISATION  421 

possess  "  anomalous  refraction  "  in  the  region  of  the  absorp- 
tion band.  For  wave  lengths  slightly  longer  than  those 
absorbed,  such  substances  have  an  exceedingly  great  refractive 
index,  and  hence  these  act  as  very  short  wave  lengths,  to 
which  the  silver  salt  is  itself  sensitive.  The  use  of  orthochro- 
matic  plates  depends  on  this  effect.  It  appears  as  if  we  are 
here  dealing  with  a  resonance  effect,  the  vibration  of  the 
electrons  of  the  dyestuff,  intensified  by  absorbing  the  light, 
causing  vibrations  of  similar  amplitude  in  the  silver  salt 
molecules.  Besides  the  organic  dyestuffs  mentioned,  it  has 
been  found  that  uranyl  salts  are  also  efficient  optical  sensi- 
tisers.  All  these  substances  fluoresce.  This,  however,  is  not 
an  essential  property.  Thus  Winther  (Zeitsch.  Wissen.  Phot., 
7,  409,  1909)  showed  that  the  light  sensitivity  of  the  Eder 
reaction,  namely,  the  reduction  of  mercuric  chloride  to  calomel 
by  means  of  ammonium  oxalate  according  to  the  equation  — 

2HgCl2  +  (NH4)2C204  =  Hg2Cl2  +  2NH4C1  +  2CO2 

which  takes  place  readily  in  the  presence  of  ferric  salt  when 
exposed  to  light,  is  due  to  the  sensitivity  of  the  non-fluorescing 
ferric  ion.  This  latter  is  reduced  in  light  to  ferrous  ion,  and 
at  the  same  time  the  oxalic  acid  is  oxidised  by  the  mercuric 
chloride.  How  this  takes  place  is,  of  course,  obscure. 

Similar  instances  of  catalytic  optical  sensitisation  have 
been  observed  by  Bruiier  (Sitzber.  d.  Krakau  Ak.  d.  Wissen.^ 
192,  1910).  Maleic  acid  is  transformed  into  fumaric  acid  in 
light,  and  this  reaction  is  optically  sensitised  by  bromine 
probably  by  the  formation  of  intermediate  compounds. 

Let  us  now  consider  another  well-known  photo-catalytic 
chemical  reaction,  namely,  the  formation  and  decomposition 
of  phosgene  — 


Of  these  three  gases,  chlorine  is  the  only  one  which  absorbs 
light  (blue)  in  the  visible  region,  and  since  this  system  has 
been  examined  in  glass  vessels  (which  exclude  any  ultra-violet) 
it  can  only  be  the  chlorine  which  acts  as  or  gives  rise  to  the 
photo-catalyst.  Working  at  500°  C  in  presence  of  light, 


422       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

Weigert  found  that  the  speed  in  both  directions  was  accelerated 
as  one  would  expert  if  an  ordinary  chemical   catalyst  were 
in   operation.      It   will   be   observed,   however,   that   in   the 
formation   of  phosgene,   the   catalysing   source,   namely,  the 
chlorine,  diminishes,  while  with  dissociation  of  the  phosgene 
the   catalyst  increases.      Weigert   looks    upon    the   catalysis 
of  the  dissociation  as  an   optical  sensitisation  whereby  the 
chlorine,  which  absorbs  the  blue,  makes  the  phosgene  like- 
wise sensitive  to  this  colour,  although  by  itself  it  would  only 
be  sensitive  to  some  of  the  much  shorter  ultra-violet  wave 
lengths.     In  a  similar  manner  Weigert  has  shown  that  chlorine 
can  be  used  as  an  optical  sensitiser  for  the  union  of  oxygen 
with  hydrogen  to  form  water,  oxygen  with  sulphur  dioxide  to 
form  sulphur  trioxide,  and  for  the  decomposition  of  ozone. 
These  reactions  will  take  place  in  the  absence  of  chlorine 
under  the  action  of  ultra-violet  light,  but  can  also  be  made  to 
take  place  even  in  the  absence  of  such  short  wave  lengths,  if 
chlorine  be  present  in  the  system  to  make  the  blue  effective. 
The  mechanism  of  such  processes  Weigert  considers  to  be  of 
the   nature   of  a   heterogeneous   catalysis.      "  The   chemical 
action  of  the  light  in  such  cases  might  be  regarded  as  consist- 
ing first  in  the  formation  of  nuclei  in  the  exposed  gases  much 
in  the  same  way  as  clouds   are  produced  in   supersaturated 
vapours.      One  may  call   these  'reaction  nuclei,'  and   their 
effectiveness  is  of  similar  nature  to  that  of  any  heterogeneous 
catalyst,   in   that    the    reacting   substances   are   adsorbed   in 
greater  density  on  the  surface,  and  in  these  places  of  higher 
concentration  the  reaction  proceeds  more  rapidly  in  accord- 
ance with  the  principle  of  mass  action."     It  will  be  observed 
that  the  reactants  themselves  are  not  assumed  to  be  photo- 
chernically  sensitive.     In  support  of  the  above  view  one  can 
cite  the  cloud  formations  in  gases  exposed  to  light,  especially 
ultra-violet  light,  the  quantitative  course  of  many  light  reactions, 
and  the  possibility  of  "  poisoning  "  by  the  addition  of  traces 
of  certain  foreign  substances. 


PHOTOCHEMICAL   REVERSIBILITY  423 

Photochemical  After-effects. 

It  has  been  found  that  in  some  cases  the  reaction  proceeds 
even  after  the  withdrawal  of  the  light.  The  phenomenon, 
which  is  called  the  photochemical  after-effect,  is  thus  roughly 
analogous  to  phosphorescence.  One  finds  this  effect  in  the 
photochemical  decomposition  of  iodoform  (Plotnikow) ;  the 
transparent  solution  of  iodoform  in  chloroform  becomes  brown 
owing  to  the  presence  of  iodine,  and  even  after  light  is  with- 
drawn the  deepening  of  colour  continues  for  several  days. 
Further,  when  a  portion  of  a  solution  which  has  been  exposed 
for  a  short  time  to  the  light  is  added  to  a  quite  fresh  solution 
the  latter  begins  also  to  decompose.  According  to  Weigert, 
the  assumption  of  heterogeneous  nuclei  formed  by  the  light 
is  sufficient  to  explain  such  after-effects  in  a  very  simple 
manner,  for  it  is  unlikely  that  such  nuclei  will  break  down 
immediately  the  light  is  withdrawn.  To  the  same  class  of 
phenomena  belongs  that  observed  by  Mellor,  namely,  that 
addition  of  exposed  chlorine  has  the  property  of  reducing  or 
removing  the  induction  period  in  the  hydrogen  and  chlorine 
combination.  Very  definite  after-effects  have  recently  been 
obtained  by  Bruner  (Bull,  de  r Ac.  des  Sciences  de  Cracovie, 
365,  1909)  in  the  photobromination  of  toluene.  The  formation 
of  the  catalyst  in  this  case  is,  however,  dependent  on  the 
presence  of  oxygen  since  the  effect  cannot  be  observed  with 
oxygen  free  solutions. 

CLASS  II. — REACTIONS  INVOLVING  A  GAIN  OF  FREE  ENERGY. 

A  characteristic  feature  of  such  reactions  is  that  on  removing 
the  light  the  system  returns  to  its  original  state.  We  thus 
have  the  phenomenon  of  photochemical  reversibility.  This,  as 
already  mentioned,  may  be  either  true  or  apparent. 

Apparently  Reversible  Reactions. 

An  example  will  make  clear  the  meaning  of  this  term. 
Let  us  take  the  case  investigated  by  Luther  and  Plotnikow 
(Zeitsch.  physik.  C/iet/i.,  61,  513,  1908),  namely,  the  oxidation 


424       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

of  phosphorous  acid  H2PO3  by  means  of  hydriodic  acid  and 
oxygen.  We  are  here  dealing  with  at  least  two  consecutive 
reactions,  the  first  of  which  is  photochemically  sensitive,  the 
second  not.  The  photo-sensitive  reaction  is  the  direct  formation 
of  iodine  from  the  hydriodic  acid  according  to  the  equation — 

aHI  +  i02  =  I2  +  H20 

This  goes  slowly  in  the  dark  and  rapidly  in  the  light.  The 
iodine  thus  produced  is  now  used  up  by  the  phosphorous 
acid,  to  reform  hydriodic  acid  and  phosphoric  acid,  according 
to  the  equation — 

H3P03  +  I2  +H20  ->  H3P04  +  2HI 

Under  the  action  of  light  the  system  becomes  brown,  due 
to  the  formation  of  iodine  (I2).  If  the  light  is  removed,  the 
solution  becomes  colourless  again,  owing  to  the  second  reaction. 
On  further  exposure  the  brown  colour  is  again  developed,  and 
may  again  be  removed  by  the  withdrawal  of  the  light.  This 
has  all  the  appearance  of  reversibility,  but  is  not  so  in  reality, 
for  gradually  the  phosphorous  acid  (H3PO3)  in  the  system  is 
being  oxidised,  and  when  this  is  complete  the  second  reaction 
will  no  longer  take  place,  i.e.  finally  a  stage  is  reached  at  which 
the  solution  which  has  been  coloured  in  the  light  is  no  longer 
discoloured  in  the  dark.  An  end  point  has  thus  been  reached, 
the  total  reaction  being  thus  represented  by — 

H3P03  -|-  i02  =  H3P04 

The  end  point  differs  from  the  initial  point,  and  the  reaction 
has  gone  in  the  natural  direction,  i.e.  the  free  energy  at  the 
end  is  less  than  at  the  beginning.  The  light  has  only  acted, 
therefore,  as  a  photo-catalyst,  and  the  process  does  not  really 
involve  a  gain  of  free  energy  at  all.  Such  reactions  belong, 
therefore,  to  the  section  already  discussed  of  those  in  which 
the  free  energy  decreases.  It  is  only  the  true  reversible 
reactions  in  which  the  free  energy  may  be  made  to  increase  by 
the  action  of  the  light. 


PHOTOCHEMICAL  REVERSIBILITY  425 

True  Reversible  Reactions. 

As  already  defined,  these  reactions  retraverse  in  the  dark 
exactly  the  same  path  they  have  traversed  in  the  light.    Under 
the  action  of  the  light  a  stationary  state  is  reached  as  has 
already  been  discussed.     It  is  of  importance  to  ask  the  ques- 
tion, What  will  be  the  effect  of  a  catalyst  on  this  stationary 
state  ?     If  this  were  an  ordinary  chemical  equilibrium,  a  simple 
catalyst  should  hasten  both  the  direct  and   reverse  reaction 
velocities  to  the  same  extent,  leaving,  therefore,  the  equilibrium 
point  unchanged.    The  stationary  state  is,  however,  not  brought 
about  by  two  opposed  mass-action  effects.     On  the  one  hand 
there  is  a  true  mass-action  effect,  which  will  manifest  itself  by 
the  system  returning  to  the  initial  state  if  the  light  be  removed. 
Opposed  to  this,  when  the  light  is  on  we  have  not  a  mass- 
action  effect,  but  an  optical  effect  of  electromagnetic  origin, 
so  that  the  direct  (optical)  and  reverse  (mass  action)  reactions 
are  of  different  origin,  and  hence  one  would  not  expect  a 
catalyst  to  affect  each  in  the  same  way.     In  other  words,  it 
is  conceivable  that  the  stationary  state  should  be  altered  by 
the  presence  of  a  catalyst.     We  find  an  illustration  of  this  in 
the  photochemical  decomposition  of  dry  and  wet  carbon  dioxide 
respectively.     Thus  under  the  action  of  ultra-violet  light  when 
the  gas  is  absolutely  dry,  almost  50  per  cent.1  of  the  carbon 
dioxide  is  decomposed  into  carbon  monoxide  and  oxygen  (the 
oxygen   being   at   the    same    time   slightly   ozonised).      The 
stationary  state  here  corresponds  to  very  large  dissociation, 
such  dissociation  being  possible  because,  as  Dixon  has  shown, 
the  mass  action  reunion   of  carbon   monoxide   and   oxygen 
scarcely  takes  place  at  all  when  the  gases  are  very  dry.     If, 
on  the  other  hand,  we  start  with  moist  carbon  dioxide  and 
expose  it  to  light,  practically  no  dissociation  can  be  observed. 
In  this  case  the  water  vapour,  which  is  the  catalyst,  has  a 
very  great  effect  on  the  carbon  monoxide-oxygen  combina- 
tion ;    so  much  so  that  the  stationary  state  is   shifted   over 
quite  to  one  side. 

1  Chapman,  Chadwick  and  Ramsbottom,  Jour.  Chem.  Soc.,  91,  942, 
1907. 


426       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

A  good  illustration  of  reversible  photochemical  reaction  in 
the  gaseous  state  &  that  of  oxygen  passing  into  ozone,  the 
equilibrium  being  shifted  by  light.  The  ionisation  of  a  gas 
by  ultra-violet  light  is  another  illustration.  Positive  and 
negatively  charged  particles  are  produced  in  equal  numbers, 
and  on  removing  the  light  these  recombine  slowly. 

Probably  the  most  important  photochemical  reaction  from 
a  technical  standpoint  is  that  of  the  reduction  of  silver  chloride 
or  other  silver  salt.  Luther  has  shown  (Zeitsch.  physik.  Chem.> 
30,  628,  1899)  tnat  *ne  pure  salt  in  the  absence  of  gelatine  or 
other  foreign  substance  decomposes  in  the  light  in  the  direction — 

2AgCl  ->  Ag2Cl  +  Cl 

and  on  removal  of  the  light  the  free  chlorine  reunites  with  the 
subsalt  according  to  the  equation — 

Ag2Cl  +  Cl  -»  AgCl 

To  a  given  light  intensity  there  corresponds  a  definite 
stationary  state.  In  connection  with  this  reaction  it  is  of 
interest  to  mention  the  observation  of  Baker  (Jour.  Chem.  Sot., 
782,  1892)  that  absolutely  dry  silver  chloride  in  presence  of 
light  does  not  darken.  Apparently  here  also  a  trace  of  water 
vapour  acts  as  a  catalyst.  The  alteration  of  the  electrical 
properties  of  solid  silver  chloride,  selenium,  tellurium,  under 
the  action  of  light  is  also  probably  an  instance  of  reversible 
photo-reaction,  in  which  the  free  energy  is  increased  by  the 
light.  Similarly,  fluorescence  and  phosphorescence  probably 
belong  to  the  same  group,  but  little  is  known  in  this  direction. 
In  these  cases  Waentig  (Zeitsch.  physik.  Chem.,  51,  436,  1905) 
has  suggested  that  under  the  action  of  light  a  new  substance  is 
formed  (even  in  very  small  quantity)  and  this  substance  is 
retransformed  into  the  original  with  emission  of  light.  This 
view  seems  to  be  supported  by  the  following  observation  on 
the  effect  of  temperature  upon  fluorescence  and  phosphor- 
escence made  by  Nichols  and  Meritt  (Phys.  Rev.,  18,  355, 
1904).  It  was  found  that  on  cooling  to  very  low  temperatures, 
fluorescent  substances  become  phosphorescent,  and  substances 
which  phosphoresce  at  ordinary  temperatures  no  longer  emit 
light.  If  we  assume  the  above  photochemical  mechanism  of 


PHOTO-DECOMPOSITION  OF   WATER    VAPOUR    427 

the  process  one  would  expect  the  reaction  to  have  a  tempera- 
ture coefficient,  and  hence  the  rapid  change  which  at  ordinary 
temperatures  produces  fluorescence  would  become  slow  at  low 
temperatures  and  proceed  even  after  the  withdrawal  of  the 
incident  light,  i.e.  fluorescence  would  become  phosphorescence. 
Similarly,  if  a  certain  speed  of  reaction  is  necessary  before  any 
light  can  be  emitted,  it  is  conceivable  that  the  slow  phos- 
phorescent reaction  might  become  too  slow  at  the  lower 
temperature  to  emit  any  light  at  all. 

The  Photo-Decomposition  of  Water  Vapour. 
(Cf.  A.  Coehn,  Ber.  d.  Deutsch.  Chem.  Gesell.,  33,  880,  1910.) 

On  exposing  a  mixture  of  electrolytic  gas  (hydrogen  and 
oxygen)  to  the  action  of  ultra-violet  light  produced  by  a 
mercury  lamp,  Coehn  showed  in  a  very  conclusive  way  that 
the  gases  combined  (the  temperature  being  about  150°  C.)  with 
one  another  "  practically  completely  "  to  form  water  vapour. 
The  extent  of  the  combination  was  so  great  that  no  de termin- 
able quantity  of  either  gas  remained.  Now,  combination  is 
the  "  natural "  direction  of  the  reaction,  and  can  be  brought 
about  by  local  heating  (i.e.  by  an  electric  spark).  It  might 
seem,  therefore,  that  the  light  only  acts  as  a  catalyst  of  a 
reaction  which  would  go  of  itself  infinitely  slowly.  This  con- 
clusion is,  however,  not  necessarily  the  correct  one,  and  it  is 
rendered  still  less  certain  by  the  behaviour  of  other  gases,  such 
as  sulphur  trioxide  dissociating  into  sulphur  dioxide  and 
oxygen  SO3  ^  SO2  +  O»  in  which  the  light  was  shown  by 
Coehn  actually  to  do  work  against  the  chemical  forces.  The 
only  way  to  settle  whether  light  can  cause  a  reaction  involving 
an  increase  in  free  energy  in  the  system  hydrogen,  oxygen, 
and  water  vapour,  is  to  find  whether  the  light  can  shift  the 
natural  equilibrium  point  reached  by  the  system  in  the  dark. 
We  may  be  able  to  decide  whether  this  is  the  case  or  not  by 
starting  with  the  system  H2O  vapour  alone  and  exposing  this 
to  light.  From  the  experiments  of  Nernst  and  von  Wartenberg 
(Zeitsch.physik.  Chem,.,  56,  543,  1906),  it  can  be  calculated  that 
at  150°  C.  the  quantity  of  hydrogen  and  oxygen  in  thermal 


428       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

or  "  dark  "  equilibrium  with  water  vapour  is  only  of  the  order 
io~30  per  cent. ;  that  is  to  say,  far  below  actual  detection  even 
by  the  most  sensitive  reagent.  The  action  of  light  may  be 
to  cause  this  dissociation  to  increase,  though  at  the  same  time 
it  is  evident  that  such  photochemical  action  would  have  to 
be  very  great  before  any  appreciable  quantity  of  either  gas 
made  its  appearance.  Even  if  negative  results  were  obtained 
it  would  still  leave  the  question  undecided,  since  there  might 
have  been  quite  a  considerable  though  still  undetectable  effect 
upon  the  equilibrium  in  such  a  case.  Coehn  investigated  this 
problem  in  the  following  way :  The  water  vapour  was  led  in 
a  slow,  steady  stream  through  a  fine  quartz  tube  close  to  the 
lamp  (in  fact,  almost  into  the  centre  of  the  light  source  itself), 
and  then  finally  into  a  eudiometer  over  mercury.  The  eudio- 
meter allowed  readings  to  be  conveniently  made  down  to 
o'oi  c.c..  After  only  half  an  hour  exposure  Coehn  found 
0-03-0-04  c.c.  of  gas  produced,  above  the  layer  of  condensed 
water  vapour,  which  proved  itself  to  be  electrolytic  gas  by 
exploding  completely  under  the  action  of  a  spark.  Ultra- 
violet light  therefore  causes  water  vapour  to  dissociate  to 
hydrogen  and  oxygen  to  an  immensely  greater  degree  than 
that  produced  by  heat  alone.  Light  must  do  work,  therefore, 
against  the  chemical  forces,  and  the  photo-dissociation  of  water 
vapour  is  a  process  involving  an  increase  in  free  energy. 
Coehn  next  proceeded  to  investigate  the  photochemical  station- 
ary state.  By  causing  the  vapour  to  pass  through  the  apparatus 
at  different  speeds,  Coehn  found  that  with  a  rate  slower  than 
that  in  which  4*35  c.c.  of  liquid  water  per  hour  collected  in 
the  eudiometer,  no  further  increase  in  electrolytic  gas  was 
produced.  The  quantity  of  gas  actually  produced  in  this 
limiting  case  was  o'88  c.c.  Using  the  results  of  several  experi- 
ments, the  calculation  of  the  degree  of  photochemical  decom- 
position of  water  vapour  (at  150°  C.),  when  the  system  has 
reached  the  stationary  state,  showed  that  this  is  at  least  0*2  per 
cent.  This  is  about  the  same  as  the  experiments  of  Nernst 
and  von  Wartenberg  have  shown  to  be  the  case  for  ordinary 
thermal  dissociation  in  the  neighbourhood  of  2000°  C.  This 
photo-decomposition,  as  Coehn  points  out,  is  probably  of 


CARBON  DIOXIDE   ASSIMILATION  429 

considerable  importance  in  meteorology,  in  connection  with 
the  behaviour  of  water  vapour  in  the  upper  layers  of  the 
atmosphere. 

Besides  gaseous  reactions,  attempts  have  been  made  to  find 
whether  electrolytic  dissociation  is  affected  by  light,  but  the 
results  have  turned  out  negative,  due  no  doubt  to  the  very 
great  velocity  of  ionic  reactions.  As  regards  organic  reactions 
which  probably  involve  an  increase  in  free  energy,  one  may 
mention  the  transformation  of  maleic  into  fumaric  acid,  and 
cinnamic  into  a-truxillic,  and,  above  all,  the  polymerisation 
of  anthracene  to  dianthracene  in  light  (the  depolymerisation 
taking  place  in  the  dark),  investigated  very  carefully  by  Luther 
and  Weigert.  This  latter  reaction  has  been  used  as  the  basis 
for  extensive  theoretical  treatment,  and  this  will  be  dealt  with 
later. 

There  are,  of  course,  many  other  photo-reactions  in  the 
domain  of  organic  chemistry,  but  they  are,  as  a  rule,  complex, 
and  in  such  cases  it  is  either  difficult  or  impossible  to  deter- 
mine whether  the  free  energy  increases  or  decreases.  One 
important  instance  of  complex  reactions  in  which  the  free 
energy  is  increased  merits  a  more  detailed  description,  on 
account  of  the  part  it  plays  in  biology,  namely,  the  assimilation 
of  carbon  dioxide  by  plants. 

This  consists  essentially  in  the  transformation  of  the  system 
(CO2  +  H2O),  which  has  very  little  free  energy,  into  the 
system  (Starch  +  O2),  which  has  a  great  deal  of  energy.  The 
total  energy  difference  (not  the  free  energy  difference)  is 
practically  that  corresponding  to  the  heat  of  combustion  of 
the  starch  and  amounts  to  685  calories  per  formula  weight 
(C6H10O5).  The  possibility  of  this  reaction  occurring  is 
intimately  connected  with  the  presence  of  the  green  colouring 
matter,  the  chlorophyll,  present  in  the  green  parts  of  the 
plant.  The  relation  of  these  substances  to  one  another  is, 
however,  very  complicated.  For  example,  if  we  take  an 
extract  of  chlorophyll  and  expose  it  to  light  the  dye  is 
bleached,  but  no  permanent  assimilation  process  goes  on. 
The  chlorophyll  does  not  act  as  a  catalyst,  for  if  this  were 
the  case  it  could  only  hasten  a  reaction  which  would  proceed 


430       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

slowly  of  itself,  i.e.  with  a  decrease  in  the  free  energy.  The 
opposite  course,  as  a  matter  of  fact,  is  followed,  the  "  natural  " 
chemical  forces  being  opposed  and  overcome,  the  free  energy 
thereby  increasing.  The  chlorophyll  must  therefore  be  used 
up  in  this  process,  the  loss  being  made  good  by  the  living 
plant.  It  is  no  longer  considered  likely  that  the  starch  is  the 
first  substance  formed ;  there  are  good  grounds  for  believing 
that  formaldehyde  is  the  first  substance  formed  and  that  it 
polymerises  to  higher  carbohydrates.  Another  view,  with  less 
experimental  justification  however,  is  that  the  first  product 
consists  of  formic  acid  and  hydrogen  peroxide  ;  whilst  a  third 
view  assumes  the  primary  formation  of  oxalic  acid  from  which 
formaldehyde  and  formic  acid  could  be  produced.  It  will  be 
seen  that  no  definite  conclusions  have  as  yet  been  reached. 

The  temperature  coefficient  of  photochemical  reactions  is 
in  general  very  small,  but  the  carbon  dioxide  assimilation 
does  not  follow  this  rule.  Thus  between  o°  and  10°  the  co- 
efficient is  2-4;  between  10°  and  20°  it  is  2*1;  and  between 
20°  and  30°  it  is  r8. 

As  regards  the  question  of  the  velocity  of  assimilation  and 
the  colour  of  the  light  *  employed,  Draper  showed  many  years 
ago  that  a  maximum  velocity  was  obtained  with  the  yellow- 
green  rays.  This  is  in  agreement  with  the  general  law  that  * 
the  rays  which  are  absorbed  are  the  chemically  active  ones. 
Draper's  observation  holds  only  for  thick  layers  in  which  the 
absorption  is  nearly  complete.  For  thin  layers  other  parts  of 
the  spectrum  play  an  important  part.  In  connection  with  this 
it  is  of  importance  to  note  the  observations  of  Engelmann 
(Bot.  Zeit2ing.,  1883-84),  who  found  that  not  only  were  the 
green  cells  of  the  plant  capable  of  the  assimilation,  but  like- 
wise the  brown  and  even  red  cells,  and  that  the  extent  of  the 
assimilation  followed  the  same  course  as  the  optical  absorption. 
Experimental  results  bearing  upon  this  were  obtained  by 
Luther  and  Forbes  in  the  quinine-chromic  acid  reaction. 
Since  the  ultra-violet  waves  were  completely  absorbed,  even  in 

1  For  a  discussion  of  "Light  Filters"  cf.  Plotnikow  (Zeitsch.  physik. 
Chem.,  79,  369,  1912);  also  C.  Winther  (Zeitsch.  Elektrochemie,  9, 
389, 


PHOTOCHEMICAL  EFFICIENCY  431 

very  thin  layers,  while  the  violet  were  only  slightly  absorbed, 
by  increasing  the  thickness  of  the  layer,  the  amount  of  ultra- 
violet absorption  would  remain  unchanged,  but  that  of  the 
violet  increased.  The  spectral  region  of  maximum  efficiency 
depends  therefore  on  the  thickness  of  the  layer,  and  this 
maximum  does  not  necessarily  correspond  with  the  region  of 
maximum  absorption  (the  ultra-violet).  In  the  green  plant 
cells  there  is  an  optical  absorption  maximum  in  the  red  and 
in  the  blue,  and  a  minimum  in  the  green,  whilst  the  yellow- 
green  (for  ordinary  thicknesses)  corresponds  to  the  maximum 
assimilation  efficiency.1 

Attempts  have  been  made  to  determine  the  efficiency  of 
photochemical  reactions  quantitatively.  By  the  efficiency  is 
meant  the  ratio  of  the  radiant  energy  turned  into  chemical 
work  to  the  total  energy  absorbed  by  the  system.  In  the 
case  of  the  carbon  dioxide  assimilation  this  problem  has  been 
very  thoroughly  investigated  by  H.  T.  Brown  and  F.  Escombe, 
Phil.  Trans.  193  B,  223,  1900;  Proc.  Roy.  Soc.,  76  B,  29,  1905. 
Brown  measured  the  quantity  of  carbon  dioxide  taken  up  by  a 
leaf  in  a  given  time,  and  supposing  it  turned  into  a  hexose,  the 
energy  transformed  is  simply  the  heat  of  combustion  (with  the 
sign  changed)  of  the  hexose  (namely,  3760  cal.-grm.);  i  c.c.  of 
carbon  dioxide  corresponded  therefore  to  5*02  calories  of 
transformed  light  energy.  It  was  observed  at  the  same  time 
that  the  assimilative  power  of  the  light  was  practically  inde- 
pendent of  the  intensity  of  the  light,  this  rather  surprising 
result  being  probably  due  to  the  fact  that  under  ordinary 
conditions  the  rays  which  were  effective  were  in  large  excess,2 
the  greater  quantity  passing  through  without  absorption  (and 
therefore  not  entering  into  the  efficiency  term  as  defined 
above).  Brown  further  observed  with  a  given  leaf  that  the 
heat  equivalent  of  the  total  light  absorbed  corresponded  to 

1  Reference  should  be  made  to  the  recent  work  of  B.  Moore,  Proc.  Roy. 
Soc.,  1914,  who  has  shown  that  iron,  especially  in  the  colloidal  form,  is 
of  great  significance  for  assimilation. 

2  A  criticism  of  Brown's  work  has  been  given  by  Tswett,   Zeitsch. 
physik.  Chem.,  76,  413,   1911,  where  the  mechanism  of  the  assimilation 
process  is  further  discussed. 


432       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

0*041  calories  per  cm.2  per  minute,  and  that  this  corresponded 
in  turn  to  the  assimilation  of  0-00034  c.c.  of  carbon  dioxide 
per  cm.'-*  per  minute.  Now  the  actual  energy  transformed  into 
chemical  work  during  the  assimilation  of  0*00034  c.c.  of  CO2 
is  0-00034  X  5 '02  cals.,  or  0-0017  cals-  Per  cni.2  per  minute, 

and  hence  the  ratio  of  -     —  or  4-1  per  cent,  is  \h&  percentage 

of  the  light  transformed,  in  other  words  the  efficiency.  When, 
however,  all  allowance  is  made  for  the  reflection  of  the  light 
at  the  surface  of  the  leaf  and  likewise  the  light  which  is  simply 
transformed  into  heat  without  doing  any  chemical  work,  it  is 
found  that  98  per  cent.,  or  practically  all  the  light  energy 
actually  absorbed  by  the  chlorophyll^  is  converted  into  chemical 
work.  For  details  of  such  calculation  the  original  papers 
must  be  consulted.  This  large  "  efficiency  "  is,  however,  not 
met  with  in  any  other  photochemical  reaction  so  far  investi- 
gated, and  is  probably  connected  in  some  way  with  the  fact 
that  the  plant  cell  is  "living."  For  the  discussion  of  the 
anthracene-dianthracene  case,  cf.  Weigert,  Ahrens  Sammlung, 
l.c.,  p.  no,  seq. 

The  term  efficiency  naturally  calls  to  mind  the  Second  Law 
of  Thermodynamics,  according  to  which  (for  any  reversible 

np »r» 

process)  the  efficiency  is  given  by  -^— — ,  where  T0  is  the 

Ao 

temperature  of  the  source  of  light  and  T  the  temperature  of 
the  reacting  system.  In  all  ordinary  cases  T  is  small  com- 
pared to  T0,  so  that  the  thermodynamic  efficiency  is  practically 
unity.  The  efficiency  of  a  photochemical  reaction  is,  however, 
evidently  not  determined  by  temperature  alone. 

We  may  conclude  this  brief  account  of  the  carbon  dioxide 
assimilation  process  by  describing  Baur's  carbon  dioxide  assimi- 
lation model.  To  appreciate  this,  however,  it  is  necessary 
to  say  something  first  about  photo-voltaic  cells  in  general. 


PHOTO-VOLTAIC  CELLS 


433 


Photo-voltaic  Cells. 

(Cf.  E.  Baur,  Zeitsch.physik.  C/iem.,  63,  683,  1908 ;  N.  Titlestad, 
Zeitsch.  physik.  Chem.^  72,  257,  1910;  E.  Baur,  Zeitsch. 
physik.  Chem.,  72,  323,  1910;  H.  Schiller,  Zeitsch.  physik. 
Chem.,  80,  641,  1912.) 

When  metals  and  other  substances  (solid  salts)  are  exposed 
to  ultra-violet  light,  it  has  been  found  that   electrons  in  the 
free   state   are   emitted   from   the    surface.      In   some   cases 
positively  charged  particles  are  also  emitted.     This  pheno- 
menon, which  has  been  fairly  thoroughly  investigated,  is  known 
as  the  "  photoelectric  effect."     This  behaviour  does  not  appear, 
however,  to  be  identical  with  the  phenomenon  which  we  are 
about  to  discuss  and  known  as  the  "  photo-voltaic  effect."     In 
the  latter  we  are  dealing  with  the  effect  of  light  upon  the 
electromotive  force  of  a  voltaic  cell,  that  is  with  the  effect  of 
light  upon  electromotive  behaviour  of  ions  in  aqueous  solution. 
Although  sufficient  work  has  not  yet  been  carried  out  to  allow 
us  definitely  to  say  what  the  mechanism  of  the  process  is — it 
may  be,  for  instance,  an  alteration,  or  tendency  towards  altera- 
tion, of    the   ordinary  ionic   equilibrium   relations, — yet   the 
phenomenon  is  a   marked  one,  and  from  its   nature   seems 
likely  ultimately   to   throw  a   great   deal  of  light   upon   the 
mechanism  of  photo-effects.     It  is  necessary,  therefore,  that 
the  student  should  make  himself  familiar  with  the  phenomenon 
even  at  this  stage.     The  sort  of  reaction  which  lends  itself 
most  readily  to  an  investigation  of  this  kind  is  that  involving 
ionic  oxidation  or  reduction.     We  shall  restrict  ourselves  to  a 
single  case,  namely,  solutions  of  uranyl-  and  uranous  salts. 
First  of  all  let  us  consider  the  following  system — 


Solution  of 
Platinum  ,  TT ,  fi  .  , 

I  uranyl  U+6  ions  and 
electrode:      urano  u+4  ions 


Saturd 
KC1 


Calomel  or 
hydrogen  half  element. 


when  unexposed  to  light.     In  a  solution  containing  metallic 
uranium  with  uranous  and  uranyl  salts,  the  equilibrium  point 
T.P.C. — ii.  2  F 


434       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

lies  over  to  the  uranous  side,  i.e.  considerable  uranous  concen- 
tration and  small  uraayl  concentration.  Unless  we  happen  to 
have  chosen  the  equilibrium  concentrations,  a  mixture  of  the 
two  salts  in  the  absence  of  the  metal  will  exhibit  an  oxidation 
or  reduction  potential  at  the  platinum  electrode.  It  will  be 
remembered  that  the  "platinum  takes  no  part  in  the  reaction,  it 
simply  serves  as  a  means  of  transferring  electrons  to  or  from 
the  solution,  when  the  system  is  set  up  in  the  form  of  a  cell 
such  as  that  indicated  above.  If  we  make  up  the  solution  so 
that  it  contains  an  extremely  large  excess  of  uranous  compared 
to  uranyl  ion,  there  will  be  a  tendency  to  form  uranyl  at  the 
expense  of  the  uranous.  That  is,  there  will  be  a  tendency  for 
electrons  to  leave  the  uranous  ions,  and  they  can  only  do  so 
by  transfer  to  the  electrode.  We  have  the  following  reaction 
tending  to  take  place  — 

—  2©->    U  +  6 


In  such  a  case  an  oxidation  process  is  tending  to  go  on  as 
regards  the  uranous-uranyl  salt  mixture.  (Naturally  if  some 
foreign  substance  were  added  which  could  be  reduced  by  the 
uranous  ions,  reduction  of  the  foreign  substance  would  take 
place  in  order  that  the  uranous  ions  might  be  transformed 
into  uranyl.  This,  however,  would  naturally  not  require  an 
electrode,  and  would  take  place  simply  in  a  test  tube,  no 
electromotive  force  being  obtained  from  it.  The  electro- 
motive force  which  one  obtained,  however,  in  the  case  under 
consideration  is  a  measure  of  the  chemical  effect  of  the 
uranous-uranyl  salt  mixture  upon  an  oxidisable  or  reducible 
foreign  substance.)  To  return  to  the  case  of  the  cell.  If  the 
reaction  — 


tends  to  occur  or  actually  does  occur  even  to  a  slight  extent, 
there  will  be  a  transfer  of  electrons  from  the  solution  to  the 
electrode.  That  is,  current  tends  to  flow  from  the  electrode 
to  the  solution  inside  the  cell.  When  the  soluti'on  contains 
an  excess  of  uranyl  ions  present  the  U+6  ions  now  tend  to 
transform  themselves  into  U+4  ions,  that  is,  a  reduction 


PHOTO-VOLTAIC  CELLS 


435 


of  uranyl  tends  to  go  on  at  the  electrode  according  to  the 
equation  — 

U+6  i 


Current  now  tends  to  flow  from  the  solution  to  the  electrode. 
Titlestad's  experiments  show  that  over  the  entire  range  of 
uranous-uranyl  mixtures  examined  by  him  the  electrode  was 
positive  with  respect  to  the  normal  hydrogen  half  element, 
that  is,  reduction  of  uranyl  was  tending  to  take  place.  The 
following  numerical  values  are  a  few  of  those  given  by 
Titlestad,  /.<:.,  and  will  illustrate  the  above  processes. 


Millimoles 
of  UOjSO,! 
per  liter. 

Millimoles 
ofU(SO4)2 
per  liter. 

Millimoles 
of  H2S04 
also  present. 

Potential  of  the 
electrode  against  the 
normal  hydrogen 
electrode  in  milli- 
volts. 

Electrolytic  poten- 
tial e  in  volts  (cal- 
culated) against 
the  hydrogen 
electrode. 

40 

12 

502 

+  0'353 

0-402 

27 

24 

502 

+  0-341 

0-404 

16 

36 

504 

+  0-32I 

0*402 

The  mean  value  of  €  obtained  from  an  extended  series  is  — 
e  =  0-404  ±  0*012  volts  (against  H2  electrode  at  25°  C.) 

It  may  be  pointed  out  that  we  have  treated  the  uranyl  ion 
UO2++  as  though  identical  with  U+6.  This  is  justifiable  —  as 
far  as  electromotive  force  values  are  concerned  —  since  there 
is  always  an  equilibrium  existing  between  the  two  owing  to 
hydrolytic  decomposition,  viz.  — 


Since,  however,  the  UO2+*  is  always  present  to  a  much  greater 
extent  than  U+6,  it  is  usual  to  write  the  reactions  with  respect 
to  uranyl.  Thus  the  oxidation  or  reduction  reaction  which 
tends  to  take  place  at  the  platinum  may  be  represented  by  — 


and  when  one  takes  into  account  the  total  reaction  in  the  cell, 


436       A   SYSTEM   OF  PHYSICAL   CHEMISTRY 

using  the  hydrogen  electrode  as  the  other  half  element,  the 
reaction  equation  is—  - 

U02++  +  4H'  +  2®  =  U+4  +  2H20 

Employing    the   usual    Peters'    formula,    we   obtain    for    the 
observed  electromotive  force  of  the  cell  the  expressions  — 

,  RTi      [UO2++][H']4 

>rr  -   f  _J  _______    IfjCT    t  __  :  ~  __  it  -       J 

2F   °S        [U+*] 


from  which  €.  is  calculated. 

Experiment  has  shown  that  the  above  is  a  truly  reversible 
oxidation-reduction  process.1  Now  we  turn  to  the  question 
of  the  behaviour  of  such  a  system  under  the  action  of  light. 
Titlestad,  l.c.t  employing  ordinary  platinum  foil  electrodes, 
set  up  a  cell  consisting  simply  of— 


Platinum 


Urano-  Urano- 

uranyl  uranyl 


Platinum 


one-half  of  which  was  exposed  to  the  light  of  a  Nernst  three- 
filament  lamp,  the  other  half  being  kept  in  the  dark.  Under 
these  conditions  the  cell  showed  quite  a  marked  and  steady  electro- 
motive  force  when  unplatinised  platinum  was  used.  Such  a 
combination  is  a  photo-voltaic  cell.  (Such  a  cell  in  the  dark 
naturally  gives  rise  to  no  electromotive  force  at  all,  since  the 
two  halves  are  identical.)  The  following  table  will  illustrate 
the  magnitude  of  the  effects  referred  to  : — 

1  Naturally  whether  a  reduction  or  oxidation  takes  place  in  the  cell 
depends  to  a  large  extent  upon  the  other  half  element  used,  since  the  nett 
direction  of  current  depends  on  the  P.D.'s  of  both  electrodes.  In  some  of 

N 
Titlestad's  experiments,  using  the  —  calomel  electrode,  oxidation  in  place 

of  reduction  actually  took  place. 


PHOTO-VOLTAIC  CELLS 


437 


Temperature,  25°  C.     Light  intensity  =  2  (Titlestad's  scale). 


Percentage  composition. 

25U+  + 

Percentage  composition. 

75U+' 

Conditions  of 
observation. 

75U+6 

25U+« 

Time  in 

E.M.F.  in 

Time  in 

E.M.F.  in 

minutes. 

millivolts. 

minutes. 

millivolts. 

0 

O 

0 

-   1-6 

Light  off 

2 

-25*4 

2 

-277 

Light  on 

4 

—  -42°6 

4 

-40-4 

,, 

12 

-67-9 

12 

-57*5 

ti 

16 

16 

24 

-72-6 

28 

-58 

n 

2 

-65 

2'5 

-48 

Light  .off 

20 

-41-1 

20-5 

-23 

,, 

40 

—  30*1 

40'S 

-H'5 

ii 

50 

-26-4 

60 

—  10 

it 

The  light  causes  the  exposed  electrode  to  become  negative 
compared  to  the  other,  i.e.  current  tends  to  flow  inside  the  cell 
from  the  exposed  to  the  unexposed  electrode. 

With  constant  light  intensity  the  final  or  maximum  values 
reached  when  exposure  has  been  continued  for  a  long  time 
are  dependent  upon  the  composition  of  the  mixture.  The 
greater  the  amount  of  uranous  salt  present  compared  to  uranyl, 
the  smaller  are  the  values  of  the  electromotive  force.  That 
is  to  say,  the  more  ^lranyl  present,  the  more  negative  is  the 
exposed  electrode.  Titlestad  confirmed  Baur's  observation, 
viz.  that  there  is  a  logarithmic  relation  between  the  intensity 
of  the  light  and  the  maximal  photo-electromotive  force.  Since 
the  maximal  photo-electromotive  force  is  a  function  of  the  total 
concentration  of  uranium  salt  as  well  as  of  the  sulphuric  acid 
content,  it  is  clear  that  the  photochemical  potential  cannot 
be  treated  as  a  function  of  temperature  and  light  intensity 
only.  In  other  words,  the  photochemical  potential  of  the 
urano-uranyl  sulphate  electrode  does  not  obey  the  law  of 
osmotic  work.  The  equation  which  holds  for  the  system  in 
the  dark,  viz. — 


438       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

cannot  therefore  be  Applied  to  the  electrode  exposed  to  light 
by  simply  adding  a  factor  dependent  upon  the  light  intensity 
alone.  This  shows  that  the  "  light  intake  "  (Lichtsaitgung)  is 
not  a  constant  for  a  given  substance,  but  depends  on  its  con- 
centration and  possibly  upon  the  presence  of  other  substances. 
The  existence  of  the  phenomenon  of  "  Positivierung"  that  is, 
the  fact  that  the  electromotive  force  after  exposure  has  ceased 
not  only  returns  to  zero  in  some  cases,  but  actually  crosses 
the  zero  point  and  sets  up  a  potential  difference  in  the  opposite 
direction  (the  electrode  now  being  positive),  shows  further  how 
complicated  this  phenomenon  is. 

Some  measurements  have  been  made  upon  pure  urano 
sulphate  solution,  and  the  careful  determinations  made  by 
Schiller  have  shown  a  slight  positive  effect  followed  by  a  large 
negative  one.1  That  is,  on  exposure  the  electrode  becomes 
first  positive  and  then  reverses,  becoming  markedly  negative 
even  as  high  as  0*3  volt.  We  must  assume,  therefore,  that 
both  uranous  and  uranyl  ions  are  photosensitive  (as  Schiller's 
experiments  with  solutions  of  the  pure  salts  have  shown), 
and  further  that  the  uranous  is  much  more  sensitive  than  the 
uranyl.  With  a  solution  of  pure  uranous  salt  exposed  to 
light  in  the  cell,  the  exposed  electrode  (ultimately)  becomes 
negative.  This  naturally  fits  in  with  the  supposition  that  the 
light  tends  to  increase  the  oxidising  tendency  (i.e.  the  transfer 
of  electrons  to  the  exposed  electrode).  What  the  mechanism 
of  the  process  is  we  do  not  know.  Baur  simply  considers  that 
the  light  alters  the  thermodynamic  potential  of  the  ions  to 
different  extents.  This  is  no  doubt  true,  but  is  too  general 
to  be  regarded  as  an  explanation.  Baur  further  considers  as 
possible  the  following  reactions  at  the  exposed  region  — 


U*  +  U6  ^>  2U5 

The  substances  on  the  right-hand  side  being  unstable,  those 

1  This  occurs  not  only  in  100  per  cent,  uranous  salt,  but  also  when 
some  uranyl  (up  to  10  per  cent.)  is  likewise  present. 


ASSIMILATION  MODEL  439 

on  the  left  must  have  their  chemical  potential  raised  by  the 
light  in  order  to  produce  them.  For  further  information 
upon  this  interesting  subject  the  papers  quoted  should  be 
referred  to. 


E.  BauSs  Carbon  Dioxide  Assimilation  Model. 

The  problem  of  the  assimilation  of  carbon  dioxide  from 
the  standpoint  of  reversible  thermodynamic  process  has  been 
approached  in  a  very  ingenious  manner  by  Baur  (Zeitsch. 
physik.  Chem.,  63,  683,  1908),  and  although  this  is  by  no  means 
the  final  solution  of  the  question,  it  is  of  importance  in  at 
least  pointing  the  way.  In  the  actual  case  carbon  dioxide  is 
by  some  means  reduced  through  the  action  of  sunlight  and  the 
chlorophyll,  the  ultimate  products  being  complex  organic  com- 
pounds. With  such  an  arrangement  as  this,  about  which  we 
know  practically  nothing,  it  would  be  hopeless  to  attempt  to 
visualise  any  reasonable  reversible  mechanism  of  the  processes 
involved.  Baur  has  therefore  confined  his  attention  to  a 
more  simple  process  of  an  analogous  nature.  The  first 
question  which  arises  is,  What  is  the  first  step  in  the  process  ? 
Baur  assumes  that  the  carbon  dioxide  is  first  converted  into 
oxalic  acid.  We  shall  accept  this  provisionally  as  correct. 
This  in  turn  probably  produces  formic  acid,  and  this  in  turn 
formaldehyde,  which  may  polymerise  into  the  carbohydrates 
and  oxyacids  which  we  know  to  be  formed  as  final  products  in 
the  plant.  Let  us  confine  our  attention  to  the  first  step  only, 
namely  the  conversion  of  carbon  dioxide  into  oxalic  acid. 
The  simplest  way  of  looking  at  this  is  to  regard  it  as  an 
electrolytic  process  taking  place  at  an  electrode  of  pure  platinum, 
the  electrode  being  necessary  as  a  transfer  of  electrons  has  to 
take  place  in  order  to  allow  the  process — 

2C02->C2O4"~+20 

to  take  place.  This  is  the  fundamental  equation  of  the  step 
as  written  by  Baur.  Since,  however,  we  do  not  as  yet 
recognise  the  physical  existence  of  positive  electrons,  the 


440       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

following  modification  of  the  equation  would  appear  to  corre- 
spond more  closely  to  an  actual  process,  viz. — 


That  is,  two  uncharged  molecules  are  supposed  to  receive  two 
electrons  (from  the  electrode),  thereby  forming  the  anion  of 
oxalic  acid.  If  we  set  up  a  cell,  one  half  being  the  calomel  or 
hydrogen  electrode,  the  other  half  element  being  a  piece  of 
platinum  dipping  into  an  aqueous  solution  of  carbon  dioxide, 
and  neutral  electrolyte,  we  should  expect  a  certain  electromo- 
tive force  corresponding  to  the  process  indicated  above,  and 
we  should  expect  some  of  the  carbon  dioxide  to  disappear,  and 
oxalanion  to  be  produced  in  the  solution.  Such  a  process, 
however,  even  in  powerful  light,  does  not  apparently  take  place 
rapidly  enough  to  allow  of  potential  measurement.  If  it  goes 
at  all  it  is  only  infinitely  slowly.  What  we  have  to  do,  there- 
fore, is  to  try  and  catalyse  it  by  making  use  of  a  rapid  ionic 
reaction  of  the  oxidation-reduction  type,  and  at  the  same  time 
make  the  light  an  essential  part  of  the  process  which,  as  a 
whole,  must  be  reversible.  A  possible  way  of  carrying  this 
out  is  effected  by  Baur's  model,  the  catalysts  employed  being 
ferrous  and  ferric  salts  1  and  silver  chloride. 

The  model  is  shown  diagrammatically  in  the  figure 
(Fig.  92).  A  vessel  is  divided  into  three  chambers,  a,  b,  c,  by 
a  photochloride  slab  and  a  semi-permeable  membrane.  The 
entire  vessel  is  filled  with  a  dilute  solution  of  hydrochloric 
acid.  The  photochloride  is  a  homogeneous  solid  solution 
consisting  of  silver  chloride  and  amorphous  silver  (cf.  K. 

1  The  use  of  iron  salts  is  naturally  suggested  by  Schafer's  observation 
(Zeitsch,  physik.  Chem.,  72,  308,  1910)  that  a  solution  of  ferrous  oxalate 
and  potassium  oxalate  in  the  presence  of  carbon  dioxide  at  5°°  C.,  and  at 
I  atmosphere  pressure,  slowly  absorbs  a  small  amount  of  carbon  dioxide, 
ferric  oxalate  being  formed.  Also  measurements  of  the  oxidation-reduction 
potential  of  ferrous  and  ferric  oxalate  ions  have  shown  that  reproducible 
values  are  obtained,  and  that  using  different  concentrations  of  ous  and  ic 
salt,  one  can  calculate  in  the  ordinary  way  the  electrolytic  potential  e, 
which  is  found  to  be  O'O2  volts  with  respect  to  the  hydrogen  electrode. 
All  these  effects  refer  to  reactions  in  the  dark.  In  the  light  Baur  has 
found  that  ferric  oxalate  solution  gives  off  carbon  dioxide. 


ASSIMILATION  MODEL 


441 


Sichling,  Zeitsch.physik.  Chem^  77,  1911 ;  also  Reinders,  Zeitsch. 
physik.  Chem.,  72,  356,  1911).  It  is  sensitive  to  light,  and 
under  its  influence  the  silver  chloride  could  react  with  the 
water  in  (a)  according  to  the  equation — 

2AgCl  +  H20  ->  2Ag  +  2HC1  +  J02 

This  has  been  experimentally  verified. 

When  this  happens  the  concentration  of  silver  in  the  photo- 
chloride  rises.  We  shall  see  the  significance  of  this  in  a 
moment.  In  compartment  c  we  have  some  ferrous  oxalate 


C0 


Light 


H20 


HCI 


HCI 


HCI 


Photo  chloride  Semi -permeable 

Membrane 

FIG.  92. 

and  ferric  oxalate  ions  (from  the  corresponding  potassium 
salt).  As  already  stated,  these  are  in  equilibrium,  the  reaction 
between  them  being — 

K3Fe(C204)3  ->  K2Fe(C204)2  +  JK2C2O4  +  CO2 

This  reaction  is  a  reversible  one.  Stating  the  above  reaction 
in  terms  of  ions  and  using  the  symbol  ^  to  indicate  that 
equilibrium  is  established,  we  may  write — 

Fe(C204)3'"  •£  Fe(C204)2"  +  JC2O4"  +  CO2  .     ( i ) 

Above  the  compartment  (<:)  there  is  a  space  containing  carbon 
dioxide,  and  the  pressure  of  the  carbon  dioxide  is  one  of  the 


442       A   SYSTEM  OF  PHYSICAL    CHEMISTRY 

determining  factors  of   the    ionic  equilibrium  referred  to  in 
equation  (i). 

The  ferric  and  ferrous  complex  ions  which  take  part  in  (i) 
will  also  further  decompose  according  to  the  equations  — 


.     .     (2) 
and  Fe(C204)2"^Fe-  +  2(C204)"   ...     (3) 


The  semi-permeable  membrane  is  supposed  to  be  permeable  to 
the  ions  Fe+++  and  Fe++  and  also  to  hydrochloric  acid.  It  is 
impermeable  to  the  complex  ions  Fe(C2O4)3'"  and  Fe(C2O4)2". 
In  chamber  (b)  we  shall  therefore  have  ferrous  and  ferric  ions, 
these  being  in  equilibrium  with  those  in  chamber  (c]  since  the 
membrane  is  quite  permeable  to  these.  The  potential  of  the 
ferri-ferro  equilibrium  is  so  arranged  that  it  is  identical  with 
that  of  the  photochloride  against  the  hydrochloric  acid  or 
Clf  in  (b). 

As  long  as  (a)  is  unexposed  to  light  the  whole  arrangement 
is  in  equilibrium.  Now  suppose  light  passes  into  (a),  as 
indicated  in  the  diagram.  The  following  reaction,  as  we  have 
already  seen,  occurs  in  (a). 

Light  +  2AgCl-t-H20->2Ag  +  2HCl+i02   .     (4) 

The  silver  produced  by  this  means  diffuses  evenly  through  the 
photochloride,  and  part  of  it  appears  at  the  other  side  of  the 
slab.  Now  at  this  (b)  side  there  was  a  potential  between 
the  photochloride  and  the  Cl'  and  the  ferrous-ferric  solution, 
the  equilibrium  having  been  altered  owing  to  the  altered  com- 
position of  the  solid  photochloride.  As  this  now  contains  too 
much  silver,  we  should  have  the  following  reaction  occurring  — 

Ag  -f  Cl'  (from  the  HC1)  +  Fe+++  ->  AgCl  +  Fe++     (5) 

Some  ferrous  ion  has  now  been  produced  in  (b)  at  the  expense 
of  the  ferric.  Diffusion  of  these  ions  in  opposite  directions 
occurs  across  the  membrane,  some  ferric  passing  into  (b)  and 
some  ferrous  passing  into  (c).  The  ferro-ferri  ion  equilibrium 
in  (c)  is  thus  upset,  there  being  now  too  much  Fe++,  and  in 
order  to  reach  the  equilibrium  again  some  of  the  ferrous 


THERMODYNAMICAL    THEORY  443 

has  to  disappear  and  give  rise  to  ferric  according  to  the 
equation  — 

Fe++  +  2C02->C204"  +  Fe+++    ...     (6) 

In  this  last  process  some  of  the  carbon  dioxide  has  been  used 
up  to  form  C2O4",  which  is  the  desired  reaction,  and  this  may 
now  take  place  rapidly  since  it  is  part  of  the  naturally  rapid 
ionic  adjustment  characteristic  of  ferrous-ferric  mixtures.  The 
nett  result  of  equations  (i)  to  (6),  as  will  be  found  by  adding 
them  together,  is  — 

2C02  +  H20  +  light  -»  H2C204 


The  photochloride  and  the  iron  salts  act  only  as  intermediaries. 

The  fact  that  each  step  of  the  process  is  of  a  reversible 
kind  allows  one  to  conclude  that  if  the  oxygen  and  oxalic  acid 
which  have  been  produced  as  above  under  the  action  of  light 
be  now  set  up  in  the  form  of  a  cell  containing  electrodes, 
reversible  with  respect  to  oxygen  and  oxalic  acid  respectively, 
we  should  receive  an  electromotive  force  from  the  cell  (in  the 
dark),  current  being  produced;  the  electric  energy  thus 
obtained  being  equivalent  to  the  light  energy  photochemically 
utilised  in  the  earlier  process.  Such  a  cell  would  be  a  photo- 
chemical accumulator. 

Baur  discusses  the  problem  at  greater  length,  but  sufficient 
has  been  said  to  indicate  the  principal  idea. 

NOTE  :  —  It  has  not  been  possible  to  find  room  for  the 
photo-phenomenon  called  Solarisation.  A  very  full  discussion 
of  the  problem  is,  however,  easily  available  to  English  readers 
in  the  papers  of  VV.  D.  Bancroft,  Jour.  Phys.  Chem.^  1909-1910. 

THERMODYNAMICS  OF  REVERSIBLE  PHOTO-REACTIONS  INVOLV- 
ING A  GAIN  IN  FREE  ENERGY  AT  THE  EXPENSE  OF  THE 
LIGHT. 

ByKs  Theory. 

This  theory  was  put  forward  by  Byk  (Zeitsch.  Elektrochemie, 
14,  460,  1908  ;  Zeitsch.  physik.  C/tem.,  62,  454,  1908).  It 
consists  in  the  first  instance  of  thermodynamical  considera- 
tions, which  of  course,  if  properly  applied,  should  hold  good 


444       A    SYSTEM   OF  PHYSICAL   CHEMISTRY 

whatever  the  actual  jnechanism  of  the  process  is,  as  long  as  it 
is  reversible.  As  regards  the  mechanism  of  the  photochemical 
action,  Byk  considers  that  it  may  be  regarded  as  essentially  a 
rapidly  alternating  electrolytic  process.  In  this  connection 
the  significance  of  the  work  of  Grotthus  has  been  brought  out 
in  the  very  comprehensive  articles  by  W.  D.  Bancroft,  four. 
Phys.  Chem.,  12,  1908,  on  the  "Electrochemistry  of  Light." 
This  can  at  least  be  regarded  as  plausible  when  one  remembers 
that  light  itself  is  an  electromagnetic  phenomenon,  Byk  starts 
with  the  assumption  that  "  no  transformation  of  material,  but 
only  of  energy ',  is  caused  by  the  intensity  of  the  incident  light, 
i.e.  the  light  energy,  and  that  the  work  performed  against 
chemical  forces  is  proportional  to  the  light  energy."  We  shall 
now  follow  out  Byk's  thermodynamical  reasoning,  taking  the 
case  of  the  reversible  reaction  in  which  work  is  gained. 

Anthracene  ^  dianthracene  in  solution 

which  has  been  investigated  by  Luther  and  Weigert.  In  the 
dark  the  natural  reaction,  i.e.  reaction  involving  decrease  of 
free  energy,  is  the  depolymerisation  process,  viz.  dianthra- 
cene ->  2  anthracene.  In  the  presence  of  light  the  reverse 
reaction,  2  anthracene  ->  dianthracene,  takes  place. 

Let  V  be  the  total  volume  of  the  solution,  [D]  and  [A]  the 
concentration  in  gram-moles  per  c.c.  of  dianthracene  and 
anthracene  present  in  the  solution,  e  the  work  done  in  trans- 
forming isothermally  and  reversibly  2  moles  of  anthracene 
into  i  mole  of  dianthracene,  supposing  each  to  be  present  at 
unit  concentration  (i  mole  per  c.c.),  which,  of  course,  is  not 
equilibrium  concentration  values.  (Note  that  this  work-term, 
being  a  thermodynamic  work-term,  does  not  mean  that  the  process 
is,  in  the  actual  case,  necessarily  brought  about  by  light.  No 
matter  how  the  change  is  brought  about,  as  long  as  it  is  done 
reversibly,  the  osmotic  work  under  the  conditions  as  regards  con- 
centration mentioned  will  be  €.)  Let  T  be  the  absolute  tempera- 
ture, and  R  the  gas  constant.  The  solution  is  considered 
sufficiently  dilute  for  the  simple  gas  laws  to  apply.  Let  k  be 
the  velocity-constant  in  the  dark  (depolymerisation  of  dian- 
thracene), /  the  time,  EA  the  total  energy  absorbed  in  unit 


THERMODYNAMICAL    THEORY  445 

time  by  the  anthracene  present  in  the  solution,  a  the  efficiency 
factor  or  fraction  of  unit  light  energy  absorbed,  which  is  turned 
into  chemical  work.  Luther  and  Weigert  carried  out  measure- 
ments of  the  absorption  of  the  anthracene  and  dianthracene  in 
phenetol  solution,  as  well  as  that  of  the  solvent  itself.  The 
extinction  coefficient  is  given  by  the  expression — 


Where  c  represents  concentration  of  the  substance  in  milli- 
moles  per  liter,  /  the  thickness  of  the  layer  traversed  by  the 
light,  I0  the  initial  intensity  of  the  light,  and  I  the  intensity 
after  traversing  the  layer  /.  Measurements  by  Luther  and 
Weigert  gave  the  following  approximate  results — 

For  phenetol  m(\  cm.)  =  o'2i 

dianthracene    w(1  cm>)  1  miuimoie  per  liter  =  o'2 1 
anthracene       ni(\  Cm.)  i  miiiimoie  per  liter  =  0*56 

This  calculation  is  made  on  the  assumption,  which  can  only  be 
an  approximate  one,  that  the  extinction-coefficient  of  a  given 
substance  is  independent  of  the  other  substances  present.  It 
will  be  noted  that  anthracene  possesses  a  much  greater  absorp- 
tion than  dianthracene.  In  the  theoretical  treatment  of  the 
subject  given  by  Byk  we  take  into  consideration  only  the  light 
absorbed  by  the  anthracene.  The  sensitivity  of  anthracene  to 
light  (as  shown  by  its  polymerising)  is  enormously  greater  than 
that  of  dianthracene ;  that  is  the  efficiency  factor  a  for  anthra- 
cene is  very  great  compared  to  that  for  dianthracene  or 
phenetol.  When  the  total  light  energy  absorbed  by  the 
anthracene  is  EA,  the  part  converted  into  photochemical  work 
is  aEA,  the  remainder  being  converted  into  heat  and  then 
dissipated. 

The  term  e. — Let  us  think  of  two  very  large  boxes,  one 
with  dianthracene  at  concentration  [D],  and  the  other  with 
anthracene  at  concentration  [A],  these  two  terms  not  represent- 
ing equilibrium  concentration.  The  question  is,  what  is  the 
work  done  by  the  system,  i.e.  by  the  chemical  forces  in  trans- 
forming i  mole  of  D  into  2  moles  of  A?  Suppose  A0 
and  D0  represent  the  equilibrium  concentration  values  of  the 


446       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

two  substances  in  the  dark.     Consider  a  large  box  with  these 
substances  present. 

first  Step.  —  Take  i  mole  of  D  from  concentration  [D] 
to  equilibrium  concentration  [D0]. 


=     -  \\df  =  \\dp  =  RT  log 


Work 

Now  let  the  molecule  of  D  [at  D0]  change  without  work  into 
2  molecules  of  A  [at  A0]  (the  principle  of  mobile  equilibrium). 
Second  Step.  —  Then  remove  these  2  molecules  of  A  from 
[A0]to[A].    ' 

Work  =  2RT  log 


[A] 
Hence  the  total  work  of  transformation  (i)  and  (2) 


We    can    write    RT  log  ^^=  -  RT  log  J2J,    which    is 
LuoJ  LAoJ 


-  RT  log  K0,  where  K0  is  the  equilibrium  constant   -pr- 

LAor 

for  the  dark  equilibrium  point,  e  has  been  defined  as  the 
work  done  in  transforming  2  moles  of  A  at  [unity],  i.e.  at 
concentration  [i],  into  i  mole  of  D  also  at  [i]  (or  the  reverse 
of  this  process  since  it  is  exactly  reversible).  Carrying  out  the 
work  process  as  before,  we  get  from  D  at  [i]  —  >  A  at  [i]  — 


=  RT  log  =  -  RT  log  K 


Hence  the  work  done  by  the  system  in  transforming  isother- 
mally  and  reversibly  i  mole  of  D  at  concentration  [D]  into 
2  moles  of  A  at  [A],  or  vice  versd,  is  numerically  — 


(simply  the  van  't  HofT  isotherm). 

When  the  transformation  takes  place  in  the  direction  of 


THERMODYNAMICAL    THEORY  447 

depolymerisation  D  ->  2  A  this  is  the  natural  way  along  which 
the  chemical  forces  act.  Again,  consider  the  solution  having 
arbitrary  concentration  terms  [D]  and  [A],  which  are  not 
equilibrium  values,  and  expose  it  to  light.  Either  anthra- 
cene or  dianthracene  will  be  formed,  depending  on  the  initial 
quantities  of  D  and  A  respectively.  Suppose  we  start  with  a 
solution  containing  a  large  quantity  of  A  and  a  small  quantity 
of  D,  D  will  be  formed,  and  the  work  equivalent  in  concentra- 
tion terms  of  the  light  absorbed  will  be,  first,  that  of  retrans- 
forming  the  amount  of  A  which  would  have  been  formed, 
during  the  time  considered  from  D,  by  the  ordinary  "  dark  " 
reaction  which  tends  to  take  place  whether  the  light  is  present 
or  not.  Secondly,  there  is  the  work  required  for  the  formation 
of  a  fresh  quantity  of  D  which  actually  takes  place.  (Note.  —  If 

,  the  relative  values  of  [D]  and  [A]  are  initially  such  that  we 
have  overstepped  the  "  light  equilibrium  "  point,  ?.<?.,  there  is 
too  much  D,  the  light  energy  absorbed  is  not  sufficient  to 
compensate  the  "  dark  "  reaction,  and  only  serves  to  slow  it 
down.  We  are  considering,  however,  for  the  sake  of  simplicity, 
the  case  in  which  D  is  actually  formed.)  As  already  seen, 

i  the  osmotic  or  thermodynamic  work  (in  sense  opposite  but 
equal  to  the  natural  chemical  work)  of  actual  formation  of 

i  i  mole  of  D  under  the  given  concentration  condition  is  — 

*  +  RTlogM 

If  dflD  is  the  newly-formed  quantity  of  D  in  gram-moles 
1  per  c.c.  in  the  time  dt^  the  work  done  in  the  solution  in  time 
[  dt  due  to  the  formation  of  D  is  — 


In  order  to  nullify  the  natural  chemical  decomposition  of 
D  which  takes  place  in  the  dark  and  happens  to  be  a  mono- 
molecular  one,  C28H20  ->  2C14H10,  the  energy  required  is  — 


1  The  velocity-constant  must  come  into  this  term  when  the  physical 
meaning  of  k  is  borne  in  mind,  namely,  the  rate  at  which  a  body  is 


448       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

The  sum  of  both  reactions  represents  the  total  work  done 
by  the  light  during  time  dt.  The  quantity  of  light  which  has 
actually  accomplished  this  is  atfL^dt.  We  obtain,  therefore— 


< 


This  gives  the  rate  of  formation  of  D  under  the  influence 
of  light.  Eyk  points  out  that  the  same  expression,  with  sign 
changed,  is  obtained  in  the  case  above  mentioned  where,  even 
under  the  influence  of  light,  the  D  decreased  (if  the  system  has 
been  arbitrarily  chosen  so  that  it  initially  has  too  much  D). 
In  the  stationary  state  (i.e.  the  photochemical  "  equilibrium  " 
point),  under  a  given  constant  light  energy  at  a  given  tempera- 
ture, the  rate  of  formation  of  D  is  just  balanced  by  the  rate  of 

decomposition,  i.e.         =  o,  and  hence  at  this  point  — 


If  we  denote  by  [Dj]  the  concentration  of  D  which  is  in 
photochemical  stationary  state  with  [A]  under  a  light  energy 
absorption  O.EA,  we  can  write  the  above  equation  — 


and  hence  equation  (i)  may  be  transformed  into  — 


decomposing  per  unit  of  time,  or  in  fact  the  amount  of  the  body  changed 
in  unit  time  if  the  concentration  of  the  body  were  kept  at  unity  throughout. 
V[D]£dt,  therefore,  is  the  mass  of  D  which  would  be  transformed  in  time 
dt  in  the  dark,  the  concentration  being  [D]  per  c.c.,  and  the  total  volume 
being  V  c.c. 


THERMODYNAMICAL    THEORY  449 

The  value  of  e  may  be  obtained  from  equation  (2)  by 
carrying  out  two  experiments  in  which  stationary  states  are 
reached,  the  absolute  values  of  A  and  D  differing  in  each  case. 
EA  has  then  different  values,  viz.  EA'  and  EA",  corresponding 
to  the  different  [A]  values.  The  same  absolute  amount  of 
light  energy  E,  to  which  the  solution  is  exposed,  is  used  in 
both  cases,  and  Luther  and  Weigert  have  given  a  formula  * 
which  is  found  to  hold  good  between  E  and  EA  corresponding 
to  differing  values  of  [A].  In  this  way  Byk  calculated  e  to 
be  72*68  X  io10  erg,  or  17,206  gram-calories,  e  represents 
the  work  done  in  the  transformation  of  i  mole  of  D  ->  2 
moles  of  A.  The  work  done  at  constant  volume  will  be 
e  —  RT  log  2,  since  RT  log  2  is  the  osmotic  work  which  has 
to  be  done  to  compress  the  system  to  the  volume  occupied  by 
the  original  material.  The  work  done  at  constant  volume  is 
therefore  16,610  gram-calories.  A  further  control  of  the  value 
of  e  may  be  obtained  by  using  it  to  calculate  the  equilibrium 
value  of  A  and  D  in  the  dark  at  equilibrium  point,  for  we  have 
already  seen  that  — 


Luther  and  Weigert  carried  out  an  experiment  in  which 
°'4785  gram  of  D  was  dissolved  in  40  grams  of  phenetol. 
After  30  hours'  boiling  in  the  dark  at  170°  C.  no  detectable 
quantity  of  D  was  present.  The  value  of  the  concentration 
of  A  (produced  from  the  D),  which  is  therefore  A0,  is 
5*507  X  io~5  gram-mole  per  c.c.  If  we  were  to  apply  the 
above  expression,  using  the  known  values  for  c  and  A0,  we 

1  The  formula  is  — 

_  m\  .  [A]  .  io6 


WA  •  [A]  .   IO6  +  mv  .  [DJ  .   IO6  +  tfZphenetol 

where  #/A>  *«D  and  w  phenetol  are  the  extinction  coefficients  of  anthracene, 
dianthracene  and  the  solvent  phenetol  respectively,  which  can  be  experi- 
mentally determined.  The  factor  io6  comes  in  Byk's  paper  owing  to 
using  gram-mole  per  c.c.  as  concentration  instead  of  millimole  per  litre 
as  employed  by  Luther  and  Weigert  (cf.  Byk,  Zeitsch.  physik.  C/iem.t 
I.e.,  p.  460). 

T.P.C.  —  II.  2  G 


450       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

should  find  that  D0=  7*175  X  io~18  gram-mole  per  c.c.,  which, 
of  course,  would  be  ^uite  undetectable,  as  experiment  shows. 

Hence  Kom°c.  is  =  r3°3  X  IQ-IS 

By  means  of  equation  (2)  and  the  value  of  e  we  can  cal- 
culate aEA. 

If  now  with  constant  light  quantity  E  absorbed  (by  solution 
as  a  whole)  and  constant  volume,  the  relation  of  the  stationary 
state  concentration  of  D  and  of  A  is  investigated,  then,  as  was 
first  observed  by  Luther  and  Weigert,  we  find  the  remarkable 
fact  that  even  when  enough  substance  is  present  to  make  the 
absorption  of  light  complete,  the  observed  values  of  the  Dj  con- 
centrations at  first  rises  (in  different  experiments  containing  in- 
creasing absolute  quantity  of  A),  but  becomes  nearly  constant 
when  the  concentration  of  the  A  is  above  200  millimoles  per 
liter.1  Luther  and  Weigert  could  only  explain  this  phenomenon 
upon  the  assumption  of  the  presence  of  a  strongly  thermal 
absorbent  intermediate  compound.  The  present  thermo- 
dynamical  considerations  alone  are  sufficient  to  predict  this 
behaviour.  Thus  let  us  write  equation  (2),  substituting  the 

1  In  the  stationary  state  produced  by  light  we  have  a  state  of  things 
differing  in  principle  from  the  equilibrium  point  reached  in  the  dark  owing 
to  ordinary  chemical  attractions  acting  according  to  the  Law  of  Mass  Action. 

Thus  in  the  dark  the  system  would  so  change  that  the  value  of  f~L  would 

LAOJ 

be  constant  whatever   the  absolute  value  of  A  or  of  D  were.     In  the 
stationary  state,  however,  such  is  not  the  case.      Even  with  the  same 

incident  light   energy  and   temperature,  the  value  of  =  —  ^  is  not  at  all 


constant,  but  varies  with  the  absolute  amounts  of  the  constituents  present. 
The  thermodynamical  theory  given  is,  however,  in  agreement  with  this, 
i.e.  it  does  not  assume  that  there  should  be  a  mass  action  equilibrium  at  the 
stationary  state.  Thus  at  first  sight  one  would  think  that  the  work  done 
in  taking  I  mole  from  the  dark  equilibrium  point  (constant  =  K0)  to  the 

light  equilibrium  point  ("K7"  —  "constant")  would  be  simply  RT  log  —  -°. 

KI 

This,  however,  has  no  meaning,  for  Kz  has  no  real  significance  as  a  constant. 
According  to  Luther,  a  real  photochemical  equilibrium  analogous  to  the 
dark  equilibrium  could  only  arise  if  the  system  at  the  same  time  had  the 
same  temperature  as  the  surrounding  "light  jacket." 


THERMODYNAMICAL    THEORY 


whole  energy  absorption  E  instead  of  EA,  which  is  nearly 
true.     Then — 


.  (  \ 

e  6 


(For  large  values  of  A  the  factor  multiplied  on  aE  becomes 
nearly  unity.)  Employing  the  above  equation,  Byk  has  cal- 
culated the  values  of  [Dj,  which  for  a  given  E  ought  to  be  in 
"  light  equilibrium  "  with  given  values  of  [A].  The  following 
table  shows  the  extent  of  the  agreement  found  :  — 


[A]  10   20   40   80    120  1  60  240  300  400 

Calculated  [D]  6*24  7*15  7-95  8-64  9'01  9-27  9'62  9-82  10-07 
Observed    [D]  372  4-59  571  7'8o  9*01  9*09  9-62  9-57    - 

(The  values  in  heavy  type  were  employed  for  the  calculation 
of  the  constants.)     The  figure  (Fig.  93)  shows  these  results 


FIG.  93. 

graphically,  Curve  i  being  the  calculated,  Curve  2  being  the 
observed  values  of  Luther  and  Weigert.  It  will  be  seen  that 
at  the  higher  range  of  A  concentration  the  agreement  between 
the  observed  and  calculated  [D]  is  good.  At  small  [A]  values 
the  difference  is  great,  and  greater  the  less  A  is  present.  This 
cannot  be  accounted  for  on  pure  thermodynamic  grounds  at  the 
same  time  taking  a  to  be  constant,  but,  according  to  Byk,  on 
electromagnetic  grounds  (in  opposition  to  Luther  and  Weigerfs 
view  of  heat-absorbing  intermediate  compounds).  An  important 
point  is  the  'practical  independence  of  the  D  concentration 


452       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

in  relation  to  the  A  concentration  for  high  values  of  the  latter, 
required  by  the  thermodynamic  theory  and  actually  obtained 
as  the  table  shows. 

The  "  light  equilibrium "  or  stationary  state  concentration 
of  D  increases  in  general  with  increasing  light  energy,  but 
there  is  always  a  lag.  This  is  seen  to  be  predicted  by  the 
theory  from  equation  (4),  which  may  be  rewritten — 


.  [A]  .  io6  -f-  mD  .  [D]  .  io6  +  Wphenetol)  =  aE    .     (5) 

If  now  E  be  increased  #-fold,  then  if  D  also  increased  «-fold, 
the  expression  on  the  left-hand  side  would  be  too  large,  i.e.  the 
equation  would  not  hold,  for  in  this  we  have  powers  of  D 

higher   than  i.     If,  therefore,  in  two  experiments   the   light 

•p> 
energy  ratio  is  =^,  then  this  is  greater  than  the  corresponding 


ratio 


In  the  following  table  are  such  values  calculated 


by  means  of  equation  (5),  and  the  observed  values  obtained 
from  Luther  and  Weigert's  data  :  — 


[A]  milliraole 
per  liter. 

_EJ 

B~ 

[El]  calculated. 

40 

I-52      I 

1-43   i 

1-46      I 

40 

37       i 

3-6      i 

40 

0*9      i 

7-8       i 

77      i 

10 

0-9       i 

7-6      i 

7'3      i 

The  agreement  is  fairly  good. 

Stationary  State  and  Temperature. 

Luther  and  Weigert  have  shown  that  as  temperature  is 
raised  the  concentration  of  D  falls  in  the  stationary  state,  other 
conditions  being  maintained  the  same,  i.e.  — 


10 


=  0*34  in  one  experiment. 


THERMODYNAMICAL    THEORY  453 

This   must   be  carefully  distinguished  from  the  temperature 

effect  upon   K0   or  r*-    !?:L  the  dark  true  equilibrium  point, 

LAoJ  J 
where  D  rises  with  rise  in  temperature. 

The  Velocity  of  Photopolymerisation. 
Considering     equation     (3),     we     see    that     the     factor 


is  very  nearly  unity,   for  €  is  a  large 


quantity  (Byk  gives  other  considerations  supporting  this  con- 
clusion :  cf.  Zeitsch.  Elektrochem.,  /.<:.,  p.  464).  We  can  there- 
fore write  — 


(6) 


is  evidently  the  velocity  of  the  true  monomolecular  dark 
reaction  of  depolymerisation.  The  term  &[D{\  should  there- 
fore approximately  represent  the  initial  velocity  of  polymerisa- 
tion, i.e.  at  the  stage  when  the  back  reaction  is  negligible. 
We  have  seen  that  for  small  A  concentrations  theory  and 
experiment  do  not  agree  very  well  in  connection  with  photo- 
equilibrium  points,  so  we  shall  pass  on  to  the  region  of  larger 
[A],  where  the  theoretical  and  experimental  curves  already 
given  lie  close  to  one  another.  In  this  region  (cf.  Fig.  93, 
already  given)  the  quantity  of  [DJ  is  almost  independent  of 
A,  so  that  we  can  regard  [D?]  as  a  constant  (£),  and  write  — 


Writing  Kz  for  kb,  we  obtain  — 

f  =  K,-*D      .....     (7) 

which  has  the  same  form  as  the  equation  found  experimentally 
by  Luther  and  Weigert  for  the  velocity  of  D  formation  in 
the  region  of  higher  A  concentration.  As  regards  the  order 
of  the  polymerisation  reaction,  Luther  and  Weigert  found  that 


454       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

it  lay  between  o  and  i.     Byk  points  out  that  this  is  to  be 
expected  from  equation  (3)  (i.e.  in  equation  (7),  if  the  term 

kD  were  negligible  the  equation        =  K?  is  evidently  of  zero 

L/£ 

order). 

The  temperature  coefficient  of  the  light  action,  i.e.  poly- 
merisation, has  been  found  by  Luther  and  Weigert  to  be  very 
small.  It  increases  with  temperature  rise  (1*1  average  increase 
per  10°  C).  Byk  shows  that  the  magnitude  of  this  effect  is 
also  to  be  expected,  say,  from  equation  (6a).  Experimentally, 

the  value  of  -        — ,  where  k  is,  of  course,  the  velocity  constant 
/&T 

[Di]T  +  10 
in  the  dark  =  2 '8  (Luther  and  Weigert) ;  also  — ,^-T- —  =  0-34 


(Luther  and  Weigert),  and  hence  for  the  product  of  the  two 
we  obtain  0*95,  while  Luther  and  Weigert  obtained  i'i. 

ByKs  Electromagnetic  Theory. 

So  far  we  have  considered  Byk's  thermodynamic  theory, 
now  we  pass  on  to  consider  the  electromagnetic  considera- 
tions which  he  brought  forward  to  complete  the  theoretical 
treatment  of  the  reversible  photo-reaction.  First  consider 
the  discrepancies  between  the  A  and  D  stationary  state  values 
for  the  region  of  small  A  (cf.  Fig.  93).  The  quantity  of 
Dz  actually  found  in  this  region  is  considerably  less  than  the 
calculated.  On  the  basis  of  foregoing  considerations  Byk 
points  out  that  this  means  that,  with  decreasing  amounts  of  A 
present,  a  larger  and  larger  fraction  of  the  light  energy  is  not 
used  for  photochemical  work,  but  for  other  purposes — in  all 
probability  for  heat.  That  is,  the  efficiency  factor  a  is  not  a 
constant,  but  depends  on  the  actual  amount  of  A  present. 
(We  should,  of  course,  expect  a  priori  the  term  aEA  to  decrease 
with  A,  but  we  have  assumed  hitherto  that  a  itself  is  constant — 
which  we  can  no  longer  consider  to  be  the  case.)  To  account 
for  this,  Byk  employs  the  following  train  of  reasoning :  It  is 
a  well-established  fact  that  the  electrical  resistance  (of  electro- 
lytes) increases  with  decreasing  concentration  and  temperature. 


BYICS  ELECTROMAGNETIC   THEORY          455 

Now  with  regard  to  continuous  current,  anthracene  solutions 
are  practically  non-conductors.  To  rapidly  alternating  current, 
such  as  light,  this  would  not  be  the  behaviour,  for  the  charged 
particles  present  in  the  molecules  could  resonate  and  take  over 
from  the  ether  (which  "  conducts  "  the  light)  a  part  of  the 
oscillating  current,  i.e.  they  would  become  conductors  for  rapidly 
alternating  current.  Now,  it  is  evidently  possible  from  this 
standpoint  that  for  sufficiently  strong  external  electromagnetic 
forces  (i.e.  great  intensity  of  light)  the  vibrating  charged  particles 
in  a  given  molecule  might  possess  such  an  amplitude  that  they 
would  be  free,  and  possibly  come  under  the  influence  of  a 
neighbouring  molecule,  which  would  thus  lead  to  the  possibility 
of  a  chemical  union  of  the  two  neighbouring  molecules,  i.e. 
photo-polymerisation.  Byk  cites  the  cases  of  the  ionising  of 
gases  and  the  photo-electric  effects  of  solid  surfaces  (shown 
actually  by  anthracene  itself),  as  well  as  the  alteration  of  con- 
ductivity of  selenium,  sulphur,  and  silver  chloride,  as  evidence 
for  the  loosening  effect  of  light  waves  upon  the  electronic 
constituents  of  various  sorts  of  molecules.  According  to  Byk, 
therefore,  we  have  the  temporary  formation  of  -j-  and  —  ions 
(very  similar  to  electrolytic  ions),  consisting  of  the  molecules 
which  have  temporarily  lost  an  electron  and  those  which  have 
gained  one.  The  actual  process  of  union  (polymerisation) 
might,  therefore,  be  rerepresented  thus — 


Now  we  come  to  an  important  conclusion.  The  fewer 
A  molecules  there  are  present  the  greater  will  have  to  be 
the  distance  traversed  by  the  charged  particles,  before  one 
such  charged  particle  can  come  under  the  attraction  of 
another  of  opposite  sign,  in  order  to  allow  of  the  possibility  of 
polymerisation.  But  the  greater  the  distance  throughout  which 
the  particles  remain  unneutralised  the  greater  the  acceleration 
which  will  be  set  up,  and  hence  a  larger  quantity  of  the  light 


456       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

energy  absorbed  (since  it  entails  friction  with  the  ether)  will 
be  converted  into  heaf,  and  the  quantity  of  light  thus  converted 
into  heat  will  be  the  greater  the  further  apart  the  A  molecules 
are,  i.e.  the  less  the  concentration  of  the  A  mojecules.  Cf. 
Byk,  Zeitsch.physik.  Chem.,  loc.  at.,  p.  482.  That  is,  on  electro- 
magnetic grounds  one  would  expect  that  with  small  A  con- 
centration the  fraction  of  the  light  (of  constant  total  E)  which 
is  available  tot  purely  chemical  work  is  less  than  in  the  case 
where  the  A  molecules  are  more  crowded.  That  is,  a  should 
decrease  with  decreasing  concentration  of  A,  which  fits  in 
with  the  observed  facts.  This  is,  of  course,  only  qualitative 
agreement.  Byk  has,  however,  endeavoured  to  treat  it  more 
quantitatively.  Thus  the  temperature  coefficient  of  the  velocity 
of  the  light  reaction  is  of  the  same  order  as  the  temperature 
coefficient  for  the  mobilities  of  the  electrolytic  ions.  The 
value  given  was  IT,  which,  however,  only  holds  for  a  certain 
narrow  range  of  anthracene  concentration  and  temperature. 
Extreme  values  are  0-98  and  1-51,  that  is  the  increase  in 
velocity  of  the  light  reaction  varies  from  o  to  50  per  cent. 
Ionic  mobilities  also  increase  from  15-27  per  cent,  for  a  rise 
of  10°  C.  The  order  of  magnitude  is  therefore  the  same  as  it 
should  be  on  the  electromagnetic  view,  a  is  therefore  a 
function  of  temperature  increasing  with  it  on  the  magnetic 
view,  since  friction  decreases  as  temperature  rises. 


Effect  of  Temperature  on  tfie  Value  of  the  D  concentration  in 
a  Stationary  State. 

Consider  equation  (2)  — 


As  temperature  rises,  a,  as  we  have  seen  from  the  electro- 
magnetic standpoint,  also  increases.  On  the  left-hand  side 
the  value  of  T  likewise  rises,  as  also  does  the  dark  velocity 
constant  k.  We  cannot  therefore  say  whether  the  ratio  of 

,      |,  i.e.  F-T       alters  or  not  ;  it  may  increase  or  decrease. 


BYK*S  ELECTROMAGNETIC   THEORY          457 

Weigert  has  shown  that  DI  diminishes  with  rise  in  tempera- 
ture. As  already  mentioned,  this  is  to  be  clearly  distinguished 
from  the  dark  equilibrium  state  for  which  Weigert  has  shown 
experimentally  that  D  is  more  stable  the  higher  the  tempera- 
ture, i.e.  he  has  shown  that  the  dark  equilibrium  constant 

\  TAW  *ncreiises  w*tn  temperature.     This  is  the  same  as  saying 


0. 

that  since — 


€  =  -  RT  log  or=  +  RT  log 


[AJ2 

€  increases  in  virtue  of  the  rise  in  T  but  decreases  in  virtue 
of  the  increase  in  D0.     Weigert  shows  that — 


C0_      4330 

'    T2 


and  therefore  that  — 


That  is- 

de  R 

—  =  —  Rlog  K0  —  -4330 

Employing  the  data  given  by  Weigert  (Ber.  D.  chem. 
Gesell.,  42,  852,  1909),  log  K0  =  —  1-82  at  85°  C.,  and  hence 
at  this  temperature  — 

=  +  .-985  X  r8,  -i:?*5 


That  is,  €  decreases  as  temperature  rises,  at  least  at  a  tempera- 
ture of  85°  C.  With  much  higher  temperatures  it  could 
become  positive.  (This  is  not  pointed  out  either  by  Byk  or 
Weigert,  but  the  temperature  is  probably  outside  the  limits  at 
which  the  solutions  could  exist  as  liquid.) 

So  far  we  have  discussed  the  joule  current  heat  (which,  if 
the  current  were  continuous  would  be  C2R)  and  likewise  the 
energy  used  for  photo-chemical  work.  Besides  these  two,  there 
is  probably  a  third  kind  of  energy  absorption  which  is  probably 
of  greater  magnitude,  namely,  the  purely  thermal  absorption 


458       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

of  the  system.  This  is  due  to  the  energy  consumed  in  the 
friction  of  the  quasi-elastic  vibrating  bound  ions  (in  contra- 
distinction to  the  free  ions,  the  friction  of  which  gives  the 
joule  electric  heat).  This  resonance  effect  determines  for 
each  single  period  of  the  external  electromotive  force  (i.e.  the 
light)  the  two  optical  constants  of  the  body,  namely,  the 
refractive  index  and  the  absorption  coefficient.  The  index  is 
connected  with  the  quasi-elastic  energy  of  the  vibrating  ions 
and  therefore  with  their  amplitude.  The  greater  the  amplitude 
of  the  resonance  vibration,  the  greater  the  probability  that  the 
molecule  will  break  up,  that  is,  a  greater  part  of  the  light  will 
be  effective  in  doing  chemical  work.  So  it  seems  that  the 
chemical  effectiveness  of  a  wave  length  will  depend  on  whether 
it  is  much  refracted  or  not.  Now  the  shortest  wave  lengths 
are  most  refracted,  i.e.  the  system  has  the  greatest  refractive 
index  for  such,  and  hence  short  wave  lengths,  violet  and  ultra- 
violet, should  be  most  photochemically  active.  This  is  a 
well  established  fact.  Further,  with  substances  exhibiting  an 
absorption  band  for  wave  lengths  just  below  the  band  (i.e.  a 
little  longer  than  those  absorbed)  refraction  occurs  to  an  ex- 
ceedingly great  extent,  such  behaviour  being  called  anomalous 
refraction.  The  dyestuffs  (optical  sensitisers)  used  to  sensitise 
plates  for  colours,  red,  yellow,  or  green,  etc.,  to  which  the 
silver  salts  are  not  sensitive  themselves,  possess  this  property 
of  anomalous  refraction,  which  fact  is  likewise  in  agreement 
with  the  principle  that  the  waves  which  are  most  highly 
refractive  are  also  most  chemically  active. 

NOTE. — The  numerical  values  quoted  in  the  above  account 
of  Byk's  theory  have  been  criticised  by  Weigert  (Zeitsch. 
Elektrochem.,  14,  470,  1908)  on  the  grounds  that  the  choice  of 
values  for  the  calculation  of  the  constants  has  been  an 
unfortunate  one.  According  to  Weigert  the  factor  a  is  really 
a  constant  independent  of  temperature,  cf.  below. 

The  Luther-  Weigert  Theory. 

Weigert  (Zeitsch.  physik.  Chem.,  63,  458,  1908)  has  put 
forward  some  thermodynamical  considerations  of  a  very 
similar  nature  to  those  of  Byk,  for  the  investigation  of  the 


THEORY  OF  LUTHER  AND    WEIGERT        459 

same  experimental  case,  the  photo-polymerisation  of  anthracene. 
After  equating  the  free  energy  change  SE  to  the  work  done  in 
isothermally  and  reversibly  decreasing  the  quantity  of  dian- 
thracene  SD  in  a  mixture  of  A  and  D,  Le.  the  natural  dark 
reaction,  Weigert  obtained — 

•• « 


whence    FT-^T  is  the  dark  equilibrium  constant  =  K0  =  • —  € 
(of  Byk). 

OT?  //TT  x/17         /// 

If  we  write  ==-  as  -—  or  as   -•-•-.  — - ,  and  remembering  that 
ou       aD  at    du 

—  is  simply  the  energy  absorbed  per  second  by  A,  i.e.  the 
aEA  of  Byk,  then  the  above  equation  becomes — 
dD 


which  is  the  same  as  Byk's  equation  (i)  without  the  term  £D. 
At  low  temperatures,  as  Byk  himself  (Ber.^  42,  1145,  J909)  nas 
pointed  out,  k  is  very  small,  and  kD  is  negligible,  so  that 
thermodynamically  the  two  theories  are  identical.  Weigert 
has  used  equation  (i)  in  the  integrated  form — 

E  =  VRTJ[D](log[g]-logK0) 

+  ([A]  +  2[D])  log  [A]  +  [D]  +  constant^  .     .     (2) 

and  since  for  all  practical  purposes  the  definite  integral  between 
the  dark  equilibrium  point  and  the  given  arbitrary  dianthracene 
concentration  is  required,  we  can  write  this  equation  in  the 
form — 


460       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

Further  (Btr.,  42,  £50,  1909),  Weigert  concludes  as  a  result 

7Tf> 

of  experiment  that  —  is  a  constant,  as  Byk  had  at  first  assumed 


to  be  the  case  for 


and  that  therefore 


and  this  is  to  be  introduced  into  the  above  expression  instead 
of  E.  As  already  mentioned,  Weigert  has  very  carefully 
measured  the  dark  equilibrium  constant  K0  for  A  ^  D  in 
toluene  at  85°  and  105°  C.,  and  it  was  found  that  raising  the 
temperature  favoured  the  existence  of  dianthracene.  In  fact, 
K0  could  be  written  as  a  function  of  the  temperature  thus  — 


log  K0  =          +  10-27 

The  following  are  some  of  Weigert's  experimental  results 
for  reactions  exposed  to  light.  Under  the  given  conditions 
0-007  calories  per  minute  was  the  chemically  active  portion 
of  the  light  energy,  and  knowing  the  value  of  K0  it  is  possible 
to  calculate  the  amount  of  D  formed  in  photo-equilibrium  by 
applying  equation  (3)  above. 


Tem- 
pera- 
ture. 

log  K0. 

Exposure 
in 
minutes. 

Initial  con- 
centration of 
anthracene. 

D  calculated 
in 
moles/liter. 

D  found  in 
moles/liter. 

65° 

-2-53 

100 

96 

I'93 

2'10 

75°) 

60 

42 

1-07 

i  -oo,  0-91,  0-92 

75° 

-2-15 

60 

97 

1-27 

1-29,  I'20,   1-24 

75U) 

100 

63 

I-84 

1-86 

8,S°) 

IOO 

44 

I  -80 

1-84,  1-82,  1-91,  174. 

-1-82 

1-85,  1-85 

»5°l 

100 

107 

2-16 

2-23,  2-19,  2-15,  2-14 

105° 

-1-18 

IOO 

43 

I-92 

1-89,  178,  i  -80,  1-98, 

1-82,  178 

105° 

-1-18 

IOO 

102 

2-31 

2-30,  2-26,  2-32,  2-25, 

2-12,  2-15 

THEORY  OF  LUTHER   AND    WEIGERT        461 

The  agreement  between  calculated  and  observed  values  of  [D] 
is  fairly  good. 

It  will  be  remembered  that  Byk  pointed  out  that  the  values 
of  [D]  and  [A]  in  the  photo-stationary  state  agreed  well  with 
that  calculated  on  thermodynamic  grounds  in  the  region  where 
[A]  was  fairly  large.  Weigert,  on  the  other  hand,  has  found 
(Zeitsch.  physik.  Chem.,  1908)  that  the  agreement  is  good  in  the 
region  of  small  [A]  values,  and  suggests  that  the  lack  of  agree- 
ment found  by  Byk  is  due  to  his  unhappy  choice  of  Weigert's 
data  for  the  calculation  of  the  constants  in  the  formula. 
Weigert  in  the  1908  paper,  therefore  sees  no  necessity  for 
introducing,  as  Byk  has  done,  an  electromagnetic  (or,  indeed, 
any  other)  explanation  for  this  discrepancy.  Whether  there 
is  a  discrepancy  or  not  must  remain  open  at  this  stage.  In 
his  1909  paper,  however,  Weigert  again  brings  in  the  inter- 
mediate compound  hypothesis.  It  should  be  remembered  in 
this  connection  that  an  important  fact,  to  which  we  have  already 
referred,  is  made  certain  by  Weigert's  experiments,  namely, 
that  the  concentration  of  D  increases  on  adding  more  A,  even 
when  the  amount  of  A  already  present  is  such  that  the  light  is 
completely  absorbed,  prior  to  the  further  addition  of  A  being 
made.  As  Byk  has  pointed  out,  the  thermodynamical  theory 
(which  was  worked  out  independently  and  practically  simul- 
taneously by  Byk  and  Weigert1)  is  not  sufficient  to  explain 
this.  Weigert  and  Luther  suggested  in  1908  (Zeitsch.  physik. 
Chem.)  63,  408  seq.y  1908)  a  chemical  explanation  of  the  fact 
that  when  the  strength  and  thickness  of  the  solution  was  such 
that  the  light  absorption  was  nearly  complete  they  found  ex- 
perimentally that,  on  further  addition  of  anthracene,  there  was 
a  still  further  increase  in  the  dianthracene  at  the  stationary 
state  concentration.  So  that,  for  example,  in  an  anthracene 
solution  containing  20  millimoles  per  liter,  the  concentration 
of  A  in  the  photo-stationary  state  was  only  50  per  cent, 
of  the  maximal  amount  (which  could  be  realised  if  a  large 

1  It  is  not  proposed  here  to  discuss  the  relative  merits  of  the  contribu- 
tions of  each  of  these  investigators  to  the  subject  of  photochemistry.  It 
is  only  fair  to  point  out,  however,  that  for  all  the  experimental  results  we 
are  indebted  to  Weigert,  either  alone  or  in  collaboration  with  Luther. 


462       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

quantity  of  anthracene  were  present),  whilst  at  the  same  time 
the  light-absorption  was  practically  100  percent  in  both  cases. 
It  is,  therefore,  necessary  to  distinguish  between  chemically 
and  thermally  absorbed  light,  for  in  the  above  instance  only 
50  per  cent,  of  the  light  was  photo-productive.  Further,  they 
brought  out  the  very  important  fact  that  the  absorption  of  the  D 
and  solvent  present  could  not  account  for  such  a  great  absorp- 
tion^ so  that  they  assume  the  existence  of  one,  or  more,  strongly 
thermally  absorbent  substances  produced  from  the  A  ?nd  which 
might  be  called  D1?  which  substance  in  turn  can  break  down 
to  give  dianthracene. 

A  rise  in  temperature,  according  to  the  Le  Chatelier-van 
't  Hoff  principle,  will  favour  the  production  of  heat-absorbing 
substances — that  is,  of  the  hypothetical  substance  referred  to — 
and  therefore  the  rate  of  production  of  dianthracene  will  be 
favoured  by  rise  of  temperature,  though  it  has  been  shown  that 
the  stationary  state  quantity  Dz  is  actually  diminished.  The 
effect  of  temperature  in  thus  altering  the  effectiveness  of  the 
light  comes  into  the  thermodynamical  expression  involving 
aEA,  which  cannot  be  regarded  as  a  constant  independent  of 
temperature  and  concentration.  We  thus  have  to  regard  the 
expression  a  as  a  variable  (either  on  the  basis  of  Byk's  electro- 
magnetic theory  or  Weigert's  intermediate  compound  theory), 
in  order  to  make  the  thermodynamic  expression  fit  the  facts. 
At  present  it  would  be  useless  to  attempt  to  decide  between 
these  two  opposing  theories  while  the  problem  is  still  only  in 
its  early  stages. 


CHAPTER    II 

Applications  of  the  unitary  theory  of  energy  (energy  quanta)  to  physical 
and  chemical  problems. 

Energy  and  Temperature  (cf.  Max  Planck,  Jour,  de  Physique^  [5] 
1,  345,  1911). — The  most  fundamental  problem  in  energetics 
is  that  of  the  relation  of  heat  energy  to  temperature.  The 
concepts  of  the  classical  theory  of  thermodynamics  which  have 
been  discussed  in  an  earlier  portion  of  this  book  have  led  us 
to  consider  temperature  as  a  factor  of  heat  energy.  Tem- 
perature bears  the  same  relation  to  heat  energy  as  force  does 
to  mechanical  work,  or  as  e^ctrical  potential  does  to  electrical 
energy.  In  fact,  any  energy  term  must  consist  of  two  parts, 
an  intensity  factor  and  a  quantity  factor.  Temperature  is 
the  intensity  factor  in  heat  energy.  The  difference  of  tem- 
perature between  two  bodies  determines  the  direction  in 
which  the  heat  exchanges  will  take  place,  just  as  mechanical 
force  determines  the  direction  of  motion,  or  the  electric 
potential  the  direction  of  the  current.  An  important  point 
in  connection  with  heat  energy  is  that  it  cannot  of  itself, 
without  the  intervention  of  external  energy,  pass  from  a  low 
to  a  high  temperature — this  being  simply  a  statement  of 
the  Second  Law  of  Thermodynamics.  Although  we  can 
learn  a  very  great  deal  about  the  relation  of  heat-rira/if/Jfir  to 
temperature  by  the  aid  of  the  Second^Law  of  Thermodynamics, 
we  find  that  the  problem  is  by  no  means  solved  completely. 
Thermodynamical  theory  tells  us  nothing  ab out  irreversible  pro- 
cesses (except  the  qualitative  relation  that  instead  of  using  the 
sign  of  equality  in  the  various  equations  with  which  we  have 
become  familiar,  we  must  simply  substitute  the  sign  of  in- 
equality, leaving  the  extent  of  the  inequality  undetermined). 
But  irreversible  processes  are,  perhaps,  the  most  important  of 


464       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

all.  Examples  of  such  processes  are  (i)  conduction  of 
heat;  (2)  radiation  *  and  (3)  diffusion.  To  gain  any  further 
information  with  respect  to  such,  we  have  to  introduce  the 
concept  of  the  kinetic  theory.  We  are  already  familiar  with 
some  of  the  outstanding  characteristics  of  the  kinetic  theory 
of  gases.  This  theory,  we  have  seen,  leads  to  the  deduction  of 
the  laws  of  Boyle  and  of  Gay-Lussac,  and  the  hypothesis  of 
Avogadro,  the  underlying  assumption  being  that  in  a  perfect  gas 
the  temperature  is  represented  by  the  mean  kinetic  energy  of  the 
molecules,  irrespective  of  the  molecular  weight.  Further,  the 
kinetic  theory  in  the  hands  of  Gibbs  and  Boltzmann  has  led, 
by  the  application  of  the  theory  of  probabilities,  to  a  generalised 
kinetic  theory  or  "  statistical  mechanics,"  in  which  the  most 
important  generalisation  reached  was  known  as  the  law  of 
equi-partition  of  kinetic  energy  among  the  various  degrees  of 
freedom  of  the  systems  considered. 

The  Principle  of  Equipartition  of  Energy. 

This  principle  was  mentioned  in  the  introductory  chapter. 
By  a  "  degree  of  freedom "  is  meant  an  independent  mode 
in  which  a  body  may  be  displaced,  or  a  possible  mode 
or  direction  of  motion.  We  shall  consider  briefly  the 
problem  of  the  number  of  degrees  of  freedom1  possessed 
by  bodies  in  the  gaseous  and  solid  states  of  matter  re- 
spectively. First  of  all  a  word  about  energy  in  general. 
We  are  familiar  with  the  two  kinds  of  energy  which  material 
systems  may  possess,  namely,  kinetic  and  potential.  These 
"  kinds  "  of  energy  are  not  to  be  confused  with  the  "  types " 
of  energy  of  which  we  are  about  to  speak.  A  "  type  "  may 
consist  of  kinetic  and  potential  energy  together,  or  simply 
kinetic  alone.  A  gas  molecule  can  possess  three  types  of  energy 
each  of  which  is  a  function  of  the  temperature  :  (i)  Energy 
of  Translation,  in  virtue  of  the  translational  motion  of  the  mole- 
cules along  free  paths  throughout  the  whole  of  the  system. 
The  energy  here  is  entirely  kinetic.  (2)  Energy  of  Vibration, 

1  The  best  account  of  statistical  mechanics  is  that  given  in  Jean's 
Dynamical  Theory  of  Gases. 


EQUIPARTITION  OF  ENERGY  465 

in  virtue  of  the  oscillations  of  the  atoms  in  a  molecule  with 
respect  to  each  other.  Vibration  of  an  atom  takes  place 
round  the  centre  of  gravity  of  the  atom.  We  might  speak  of 
it,  therefore,  as  circular  vibration.  The  energy  of  vibration 
is  partly  kinetic,  partly  potential.  The  possibility  of  a  gas 
molecule  possessing  both  types  of  energy  depends  on  whether 
the  molecule  is  monatomic  or  polyatomic.  The  energy  of 
translation  belongs  to  any  gas  molecule,  and  is  independent  of 
the  number  of  atoms  in  the  molecule.  Since  translation  can 
take  place  along  each  of  the  three  space  co-ordinates  X,  Y,  Z, 
a  gas  molecule  possesses  three  degrees  of  freedom  in  virtue  of 
this  translational  motion.  As  regards  the  number  of  degrees 
of  freedom  in  virtue  of  vibration^  it  is  assumed — and  experi- 
ment bears  it  out — that  a  monatomic  gas  molecule  possesses 
zero  degrees  of  freedom,  since  it  does  not  vibrate  about  a 
centre  of  gravity,  but  the  molecule  or  atom  is  simply  trans- 
lated from  place  to  place.  Such  a  molecule  seems  to  function  as 
a  movable  massive  point.  \Note. — Perhaps  we  should  consider 
that  a  collision  with  another  molecule  involves  vibration,  but 
this  question  has  not  been  settled.]  A  diatomic  molecule,  on 
the  other  hand,  consisting  of  two  massive  points,  held  at  a  fixed 
distance  apart,  possesses  one  degree  of  vibrational  freedom  in 
virtue  of  motion  of  approach  or  retreat  of  the  two  atoms  along 
the  line  joining  them.  In  a  triatomic  molecule  the  atoms 
A,  B,  C  probably  vibrate  in  pairs,  AB,  BC,  CA,  each  pair 
functioning  like  a  diatomic  molecule,  so  that  the  molecule  has 
three  degrees  of  freedom  in  respect  of  vibration.  When  this 
is  added  to  the  three  degrees  in  virtue  of  translation,  we  obtain 
six  degrees  for  the  two  types  in  the  case  of  a  triatomic  molecule. 
(3)  The  third  type  of  energy  is  Energy  of  Rotation,  in  which 
the  molecule  as  a  whole  rotates,  such  rotation  being  with 
respect  to  the  three  dimensions  of  space.  Monatomic  gas  mole- 
cules appear,  however,  to  be  without  rotational  energy,  just 
as  they  are  without  vibrational  energy.  This  conclusion  rests 
on  the  experimental  fact  that  the  specific  heat  of  argon  and 
other  monatomic  gases  and  metallic  vapours  can  be  accounted 
for  by  simply  assuming  translational  energy,  as  we  shall  show 
later  on.  A  monatomic  gas  molecule  appears  to  function 
T.P.C. — n.  2  H 


466       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

simply  as  a  massive  point,  and  not  as  a  massive  sphere  When 
a  molecule  consists  of  more  than  one  atom,  however,  it  seems 
reasonable  to  ascribe  to  it  a  certain  amount  of  rotational 
energy.  One  would  be  inclined  to  think  that  this  should  be 
distributed  along  three  degrees  of  freedom — as  in  the  case  of 
the  sphere — no  matter  how  many  atoms  were  present  in  the 
molecule,  for  in  rotation  the  molecule  acts  as  a  whole.  Pro- 
bably a  ^'atomic  molecule  does  not  possess  three  degrees  of 
freedom,  but  only  two  in  respect  of  rotation,  since  the  spatial 
arrangement  of  the  atom  is  necessarily  linear.  No  general 
agreement  has  been  reached  on  the  question. 

Reference  may  be  made,  however,  to  a  paper  by  N.  Bjerrum 
(Zeitsch.  Elektrochem.)  17,  731,  1911 ;  also  ibid.,  18,  101,  1912), 
who  considers  mono-  di-  and  triatomic  gas  molecules  to  possess 
the  following  number  of  degrees  of  freedom,  and  the  correct- 
ness of  these  assigned  values  is  to  a  certain  extent  borne  out 
by  the  values  for  the  atomic  heat  of  the  respective  gases. 


No.  of  atoms  in 
the  molecule. 

Number  of  degrees  of  freedom. 

In  virtue  of 
translation. 

In  virtue  of 
rotation. 

In  virtue  of 
vibration. 

I 
2 

3 
4 

3 
3 
3 
3 

0 
2 

3 

3 

O 

I 

1 

Our  knowledge  of  the  liquid  state  is  so  scanty  that  it  has 
not  been  possible  hitherto  to  assign  any  definite  value  to 
possible  number  of  degrees  of  freedom.  A  liquid  molecule 
must,  of  course,  possess  three  degrees  of  freedom  in  respect 
of  translation  ;  it  must  likewise  possess  an  unknown  number 
in  respect  of  rotation  and  vibration.  We  pass  on,  therefore, 
to  the  solid  state.  In  the  case  of  a  solid,  i.e.  a  crystalline 
substance,  and  probably  also  in  the  case  of  a  super-cooled 
liquid  like  glass,  it  is  customary  to  regard  translational  energy 
(and  also  rotational  energy)  as  absent,  the  energy  possessed, 
by  the  molecules  of  a  solid  being  vibrational.  In  the  case  of 


EQUIPARTIT10N  OF  ENERGY  467 

monatomic  solids,  e.g.  metals,  vibration  of  each  atom  can  take 
place  along  all  three  axes,  so  that  even  the  simplest  type  of 
solid  possesses  three  degrees  of  freedom.  We  shall  return  to 
this  later. 

For  the  present  we  have  to  take  up  the  application  of  the 
principle  of  equi-partition  of  (kinetic)  energy  amongst  degrees 
of  freedom.  According  to  this  principle,  when  a  system  is  in 
statistical1  equilibrium^  such  equilibrium  being  determined  by 
a  number  of  variables,  i.e.  degrees  of  freedom,  to  each  such 
variable  one  must  attribute  the  same  qitantity  of  (kinetic)  energy. 
It  is  to  be  remembered  that  there  is  no  restriction  as  regards 
the  physical  state  of  the  system  —  the  principle  applies  equally 
well  to  gaseous,  liquid  or  solid  systems  and  to  systems  em- 
bracing two  or  more  such  states  simultaneously.  It  is  sup- 
posed to  hold  equally  well  also  for  degrees  of  freedom  in 
respect  of  translation,  vibration  or  rotation.  To  see  how  much 
this  energy  amounts  to  per  degree  of  freedom,  let  us  consider 
the  kinetic  energy  of  translation  of  a  perfect  gas  at  a  given 
temperature.  First  of  all  we  have  the  relation  — 

PV  =  RT 

Further,  we  have  seen  in  the  theoretical  deduction  of  Boyle's 
Law  that  — 


where  p  is  the  density  of  the  gas  and  u  the  root-mean-square 
velocity.     If  we  are  considering  i  gram-molecule  of  the  gas, 

1  When  a  system  consists  of  a  large  number  of  individuals  the  only 
feasible  method  of  studying  the  behaviour  of  the  system  is  by  applying 
statistical  means.  That  is,  we  do  not  attempt  to  follow  out  the  extremely 
complicated  path  followed  by  each  single  individual,  but  instead  we  apply 
the  principle  of  probability  —  as  regards  distribution  of  energy,  for  example 
—  to  the  system  as  a  whole.  In  this  way  Maxwell  deduced  his  law 
regarding  the  distribution  of  velocities  amongst  the  molecules  of  a  gas, 
showing  that  while  all  velocities  were  possible  only  a  very  few  molecules 
possessed  either  a  very  great  or  a  very  small  velocity,  the  greater  propor- 
tion possessing  velocities  lying  within  relatively  narrow  limits.  This  is 
the  equilibrium  distribution  of  molecular  velocities.  In  fact,  if  all  the 
molecules  were  given  the  same  velocity  it  can  be  shown  that  this  would 
only  be  a  temporary  state,  the  system  eventually  reaching  the  statistical 
equilibrium  denned  by  Maxwell's  Law. 


468       A   SYSTEM  OF  PHYSICAL    CHEMISTRY 

molecular  weight  M 
p=  v     ^-a — ,  where  V  is  the  molecular  volume,  and 

hence  we  can  write — 

PV  =  £M«2  =  lNw«2 

where  m  is  the  mass  of  a  single  molecule  and  N  the  number 
of  molecules  in  i  gram-mole.  Taking  into  the  consideration 
the  first  equation,  we  obtain — 

RT  =  iM«2  =  lN/w/2 

o  a 

Now  the  kinetic  energy  of  a  single  molecule  is  J;;/«2j  an(j 
hence  the  kinetic  energy  of  one  gram-mole  is  ^Nmu2.  That  is 
the  kinetic  energy  of  i  gram-mole  is  |RT,  and  this  holds  good 
whether  the  gas  be  monatomic  or  polyatomic.  Now  a  gas 
molecule  has  three  degrees  of  freedom  in  virtue  of  translation, 
and  hence  by  applying  the  equipartition  principle,  EACH  DE- 
GREE OF  FREEDOM  POSSESSES  ENERGY  EQUAL  TO  £RT. 

Let  us  consider  the  special  case  of  a  monatomic  gas.  As 
we  have  already  seen,  it  can  possess  no  rotational  or  vibrational 
energy.  Its  energy,  due  to  translational  motion,  for  i  gram- 
mole  at  T°  abs.  is  |RT,  when  R  =  1*985  cals.  Now  the  specific 
heat  Cv  at  constant  volume  is  simply  the  increase  in  the  total 
energy  per  degree.  If  we  are  dealing  with  i  gram-mole  or 
gram  atom  as  unit  of  mass,  the  heat  term  will  be  the  so-called 
molecular  or  atomic  heat  Cv>  It  is  clear  from  definition  that 
the  increase  in  the  (kinetic)  energy  of  translation  i  per  i°  rise 

in    temperature  ==™  (f  RT)  ==|R  ==  2*98     cals.    per     mole. 

Recent  experiments  of  Pier  in  Nernst's  Laboratory  (Zeitsch. 
Elektroctiem.,  15,  546, 1909 ;  16,  897,  1910)  have  shown  that  the 
molecular  heat  of  argon  is  2*98  cals.  per  mole,  and  further 
that  this  is  independent  of  temperature.  This  is  in  complete 
agreement  with  the  above  value  calculated  on  the  basis  of  the 
equipartition  principle  as  applied  to  translation.  It  suggests 
that  one  should  neglect  the  rotational  energy  of  the  monatomic 
molecule  (vibration  of  atoms  inside  the  molecule  is  naturally 

1  Energy  of  translation  is  necessarily  entirely  kinetic,  and  it  is  to  the 
distribution  of  kinetic  energy  amongst  degrees  of  freedom  that  the  law  of 
equipartition  is  properly  to  be  applied. 


EQUIP ARTITION  OF  ENERGY  469 

impossible  since  the  gas  is  monatomic).  In  fact  the  mon- 
atomic  molecule  seems  to  function  as  a  massive  point.  Agree- 
ment of  this  order  between  calculated  and  experimental  values 
is,  however,  not  found  in  other  cases.  Thus,  taking  the  case  of 
a  diatomic  gas,  the  number  of  degrees  of  freedom  in  virtue  of 
translation  is  again  3.  The  number  of  degrees  of  freedom 
in  virtue  of  vibration  we  have  considered  as  i ;  that  is  4 
degrees  in  virtue  of  translation  and  vibration.  The  corre- 
sponding kinetic  energy  of  such  a  molecule  will  be  4  X  JRT 
=  4*0  T  cal./mole.  Of  course  this  does  not  represent  all  the 
energy  due  to  translation  and  vibration.  In  vibrations  we 
have  potential  energy  as  well  as  kinetic  which  must  be  taken 
account  of.  It  can  be  shown  by  a  simple  calculation  that  the 
potential  energy  of  a  particle  undergoing  what  we  might  call 
"  circular  vibration  "  is  just  equal  to  the  kinetic  energy  of  the 
vibration.  The  calculation  is  as  follows  : — 

Consider  a  vibrating  particle  whose  amplitude  of  vibration 
is  so  small  that  we  can  consider  that  the  force  tending  to  draw 
it  back  to  its  centre  of  gravity  (position  of  rest)  is  proportional 
to  the  distance  of  the  particle  from  this  centre.  Suppose  that 
r  is  the  radius  of  the  imaginary  circle  traversed  by  the  particle 
with  velocity  n.  If  m  is  the  mass  of  the  particle,  we  can 
equate  the  so-called  centrifugal  and  centripetal  forces,  viz.— 

wu% 

—  =  A.r 
r 

where  A  is  the  force  with  which  the  particle  is  drawn  in 
towards  the  centre  when  the  particle  is  unit  distance  from  the 
centre.  Since  the  particle  is  at  distance  r  from  its  centre  of 
gravity,  we  likewise  know  the  value  of  its  potential  energy, 
namely— 


/: 


This  expression  is  identical  with  \muz9  as  is  seen  by  com- 
parison with  the  above  equation.  But  the  term  \muz  is  the 
average  kinetic  energy  of  the  vibrating  particle,  and  hence  in 
a  complete  vibration  the  kinetic  energy  is  just  equal  in  magni- 
tude to  the  potential  energy. 


470       A   SYSTEM   OF  PHYSICAL   CHEMISTRY 

,  Now  a  circular  ^ibration  of  the  sort  considered  involves 
one  degree  of  freedom,  i.e.  the  line  of  junction  of  the  two 
atoms.  On  the  equipartition  principle  the  kinetic  energy 
involved  per  mole  is  JRT.  Since  there  is  likewise  an  equal 
amount  of  potential  energy,  the  total  energy  due  to  vibration 
is  iRT.  Adding  the  amount  due  to  the  (kinetic)  energy  of 
translation  of  the  diatomic  molecule  as  a  whole,  viz.  f  RT,  we 
obtain  |RT  as  the  total  energy  due  to  translation  and  vibration. 
That  is,  the  rise  in  this  energy  per  i°  is  £R  =  5'o  cal.  per 
mole  per  degree,  taking  R  —  2  cals.  If  now  we  take  rotation 
of  the  molecule  as  a  whole  into  account  we  again  have  two 
degrees  of  freedom,  to  which  one  must  assign  RT  units  of 
kinetic  energy.1  The  total  energy  of  a  diatomic  molecule, 
provided  the  law  of  equipartition  is  true,  and,  provided  ALL 
the  degrees  of  freedom  are  effective^  should  be  f  RT,  and  the 
molecular  heat  therefore  JR  =  7  *o  approx. 

Experiment  shows,  however,  very  different  values.  For 
hydrogen  at  o°  C.  the  molecular  heat  C«  =  4*9  to  5*2  cal.  per 
degree,  and  at  2000°  C.,  Cu  =  6'5  cal.  (cf.  Nernst,  Zeitsch. 
Elektrochem.)  17,  272,  1911).  For  nitrogen  at  o°  Cr  =  4*84, 
and  at  2000°  Cv  =  6*7.  For  chlorine  at  o°  Cv  =  5*85,  at 
1200°  Ct>  =  7*0.  For  oxygen  at  o°  C«  =  4*9,  at  2000° 
Cy  =  6-7.  Not  only  is  there  lack  of  agreement  in  the 
numerical  values  between  those  observed  and  those  calcu- 
lated at  lower  temperatures,  but  the  fact  that  the  molecular 
heat  of  diatomic  gases  varies  considerably  with  the  temperature 
is  quite  unaccounted  for  by  the  theory  of  equipartition  unless, 
indeed,  the  number  of  degrees  of  freedom  is  a  function  of 
the  temperature.  We  shall  refer  briefly  to  this  later. 

In  the  case  of  triatomic  gas  molecules  the  degrees  of 
freedom  in  respect  of  translation  are  three,  the  [kinetic]  energy 
corresponding  being  |RT.  As  regards  vibration,  there  are 
possibly  three  vibrating  pairs  each  with  one  degree  of  freedom, 

1  The  concomitant  potential  energy  in  rotation  is  shown  by  Bjerrum, 
loc.  cit.t  to  be  a  negligible  quantity.  As  regards  rotation  therefore  it 
appears  that  we  should  regard  a  diatomic  molecule  as  a  cylinder,  possessing 
consequently  two  axes  for  rotation,  i.e.  2  degrees  of  freedom  as  far  as 
rotation  (of  the  molecule  as  a  whole)  is  concerned. 


EQUIPARTITION  OF  ENERGY  471 

corresponding  to  the  quantity  3  X  |RT  of  kinetic  energy.  To 
this  has  to  be  added  an  equivalent  amount  of  potential  energy  f 
making  3RT  as  the  total  energy  term  in  respect  of  vibration. 
Hence  translation  and  vibration  apparently  entail  |RT  units  of 
energy,  and  the  increase  in  this  for  i°  rise  in  temperature  is 
f  R  =  9*0  cal.  per  mole  per  degree.  Logically  we  should  like- 
wise add  a  term  for  rotation  of  the  molecule  as  a  whole,  which 
we  have  seen  amounts  to  |RT.  The  observed  molecular  heat 
Cy  for  CC>2  at  18°  is  7*09,  and  this  becomes  10*47  at  2210° 
(Pier,  I.e.}.  For  water  vapour  C«  at  50°  C.  =  5-96  (Nernst 
and  Levy),  and  at  2327°  C.,  Cu  =  9-68  (Pier).  Again  the 
discordance  between  theory  and  experiment  is  very  apparent 
since  theory  predicts  a  constant  molecular  heat  of  12  cals.  per 
mole  per  degree. 


TIic  ratio  y  for  Gases,  and  the  number  of  "  effective"  Degrees 
of  Freedom. 

It  may  easily  be   shown   (cf.  Meyer's  Kinetic   Theory  of 

r* 

Gases,  p.   140  seq.)  that  the  ratio  ~-  or  y  can  be  expressed 
approximately  in  the  form  — 


where  //  is  the  number  of  degrees  of  freedom  of  the  gas 
molecule.  In  the  case  of  monatomic  gases  the  value  of  y  is 
1*666,  and  this  is  the  quantity  which  is  obtained  on  putting 
//  =  3  in  the  above  expression.  That  is,  a  monatomic  gas 
molecule  possesses  three  degrees  of  freedom,  in  respect  of 
translation  only.  This  agrees  with  the  conclusion  we  came 
to  above  (though,  of  course,  it  must  be  remembered  that  the 
actual  number  of  degrees  is  possibly  greater  than  this,  i.e. 
degrees  in  virtue  of  rotation,  but  for  some  unknown  reason 
only  some  are  effective  in  regard  to  heat  capacity,  for  after 
all  an  atom  is  not  a  point  in  the  mathematical  sense).  In  the 
case  of  a  diatomic  molecule  the  value  of  y,  found  by  experi- 
ment, is  in  many  cases  1*4,  and  this  will  correspond  to  putting 


472        A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

n  =  5  in  the  above  equation.  In  triatomic  gas  molecules 
y=i'3  in  general,*and  this  makes  n  =  7.  These  numbers 
are  certainly  less  than  the  actual  number  of  degrees  of  freedom 
possessed  by  di-  and  triatomic  gases,  as  is  indeed  shown  by 
the  fact  that  there  are  some  diatomic  gases  with  as  small  a 
value  as  1*29  for  y  (iodine  vapour),  and  some  triatomic  gases, 
e.g.  CS2,  for  which  y  =  1*2.  Further  the  values  of  y  are  not 
constant  but  vary  with  the  temperature.  The  whole  problem 
of  the  number  of  degrees  of  freedom  is,  therefore,  in  a  very 
unfinished  state. 

The  doubt  which  exists  in  the  case  of  polyatomic  mole- 
cules, regarding  the  true  number  of  degrees  of  freedom,  takes 
away  from  the  force  of  the  criticism  levelled  against  the 
principle  of  the  equipartition,  on  the  ground  of  the  lack  of 
agreement  between  observed  and  calculated  molecular  heats. 
The  soundest  criticism  of  the  principle  rests  on  the  experi- 
mental observation  that  the  molecular  heat  varies  with  the 
temperature,  whilst  the  principle  leads  us  to  expect  it  to 
be  constant,  no  matter  how  many  degrees  of  freedom  be 
present. 

Since  the  equipartition  principle  is  considered  to  hold 
equally  well  for  all  states  of  matter,  the  term  JRT  must 
represent  the  kinetic  energy  per  degree  of  freedom  of  i 
gram-mole  of  any  system  at  the  temperature  T.  In  solid 
elements — the  metals — it  is  generally  agreed  that  the  gram- 
molecule  and  gram-atom  are  identical,  so  that  in  such  a  case 
JRT  is  the  kinetic  energy  per  degree  of  freedom  for  each 
gram-atom,  R  being  put  equal  to  1*985  abs.  How  many 
degrees  of  freedom  does  an  atom  in  a  solid  such  as  a  metal 
possess  ?  It  is  customary  to  regard  the  motions  of  the  atom 
in  a  monatomic  crystalline  solid  as  taking  the  form  of  vibrations 
round  a  fixed  point  or  centre  of  gravity.  The  existence  of 
the  crystalline  form  seems  to  preclude  free  translation  as  in  the 
case  of  gas  molecules.  Further,  the  absence  of  rotation  of 
monatomic  molecules  in  gases  suggests  a  similar  absence  of 
rotation  of  metallic  atoms,  at  least  when  the  temperature  is  not 
too  high.  The  orientation  of  the  atom  of  a  metal  is  such,  that 
it  is  free  to  vibrate  in  any  direction  which  can  be  resolved 


EQUIPARTITION  OF  ENERGY  473 

in  terms  of  the  three  space  co-ordinates  X,  Y,  Z.     Such  a 
particle  possesses  three  degrees  of  freedom.1 

The  kinetic  energy  in  virtue  of  this  "  space  "  vibration  is 
evidently  |RT,  and  since  any  vibration  involves  an  amount  of 
potential  energy  equivalent  to  the  kinetic,  the  total  energy 
of  i  gram-atom  of  a  monatomic  solid  is  sRT,  according  to 
the  principle  of  equipartition  of  energy.  Now  the  atomic 
heat  C«  at  constant  volume  is  simply  the  change  of  total  atomic 
energy  per  degree,  that  is — 

C,  =  ~  (3RT)  =  3R  =  5*955  cals. 

The  application  of  the  equipartition  principle  has  therefore  led 
to  tfie  conclusion  that' the  atomic  heat  of  monatomic  solids  should 
be  a  constant,  viz.  5*955,  the  same  for  all  monatomic  solids  and 
independent  of  temperature  (the  term  $R  does  not  contain  T ; 
though,  of  course,  the  total  energy  present  in  the  solid  at  any 
given  temperature  depends  on  this  temperature,  viz.  3RT  in  the 
above  case).  This  result  is  practically  identical  with  Dulong 
and  Petit's  experimentally  discovered  law,  and  this  agreement 
is  one  of  the  most  striking  pieces  of  evidence  in  favour  of  the 
equipartition  principle.  The  following  table  gives  some  of 
the  values  of  atomic  heats  determined  by  experiment.  They 
are  in  general  mean  values  holding  for  the  range  0-100°  C. 

1  In  the  case  of  a  diatomic  gas  molecule  the  relative  vibration  of  one 
atom  with  respect  to  the  other  was  only  considered  to  possess  one  degree 
of  freedom.  (This  as  a  matter  of  fact  is  a  disputed  point.)  In  the  case  of 
metals  each  atom  vibrates  independently  of  others,  at  least  as  long  as  its 
amplitude  is  less  than  the  mean  distance  of  the  atoms  apart,  and  hence 
three  degrees  of  freedom  must  be  ascribed  to  it. 


474       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

3R  =  5'955- 


Element. 

Atomic  heat. 

Element. 

Atomic  heat. 

Li 

6-6 

La  and  Ce 

6-2 

Na 

67 

W 

6-1 

Mg 

5  '9 

Ir. 

6'2 

Al 

5  '9 

Au 

6-2 

P 

6-2 

Pb 

6-4 

S 

5'9 

Th 

6-4 

K 

6'5 

Pel 

6'3 

Ca 

67 

Ag 

6-0 

Ti 

5*4 

Cd 

6'i 

Cr 

6'3 

Su 

6'5 

Mn 

67 

Sb 

5'9 

Fe 

6-4 

Cs 

6-4 

Co 

6-0 

Ta 

6-6 

Ni 
Cu 

6'3 

5'9 

Os 
Pt 

5'9 
6-2 

Zn 

Tl 

6-6 

As 

6-2 

Bi 

6-3 

Se 

6-6 

Ur 

6-6 

Rh 

5*9 

1 

I 

The  approximation  of  these  values  to  that  of  3R  is 
obviously  very  close.  Nevertheless,  too  great  stress  cannot  be 
laid  upon  this,  for  it  has  been  shown  by  experiment  that  the  atomic 
Jieat  of  solids  (as  in  t/iecase  of  gases)  is  a  function  of  the  temper a- 
ture,  decreasing  to  very  small  values  at  very  low  temperatures. 
We  shall  take  this  question  up  later  on  in  discussing  the  work 
of  Nernst  and  his  collaborators.  Sufficient  has  been  said, 
however,  in  this  connection  to  show  that  while  the  principle  of 
cquipartition  is  partially  true,  it  is  not  in  the  form  given  by 
Boltzmann  sufficiently  comprehensive.  It  might  be  thought 
that  a  sufficient  explanation  of  the  observed  increase  in  atomic 
heat  with  rise  in  temperature  lies  in  the  supposition  of  new 
degrees  of  freedom  coming  into  existence.  We  cannot,  how- 
ever, imagine  a  "fractional"  degree  of  freedom.  It  must 
either  exist  definitely,  or  not  at  all.  One  would  expect,  there- 
fore, that  the  atomic  heat  should  rise  by  steps  as  the  tempera- 
ture rises.  All  observations,  however,  have  shown  that  the 
increase  in  atomic  heat  is  a  perfectly  continuous  function  of 
the  temperature.  Leaving  the  problem  of  specific  or  atomic 


THE   QUANTUM   THEORY  475 

heats,  let  us  turn  to  another  important  problem,  namely,  that 
of  thermal  radiation ;  for  it  was  through  investigation  carried 
out  in  this  field  that  the  modifications  of  the  principle  of  equi- 
partition  were  eventually  introduced,  which  in  the  hands  of 
Planck  and  Einstein  have  permitted  a  satisfactory  explana- 
tion to  be  given  of  the  discrepancies  hitherto  existing  between 
theory  and  experiment,  not  only  in  the  domain  of  radiation 
itself,  but  likewise  in  that  of  the  heat  content  of  solids. 
Whether  these  modified  views  form  the  ultimate  solution  of 
the  problem,  it  is  at  present  impossible  to  say.  They 
represent,  at  any  rate,  a  fundamental  stage  in  the  develop- 
ment of  the  subject. 

PLANCK'S  THEORY  OF  RADIATION.* 

In  studying  the  question  of  radiation,  that  is  of  the  exchange 
of  radiant  energy  between  matter  and  ether,  it  is  necessary,  of 
course,  to  limit  our  consideration  to  the  equilibrium  state.  If 
an  enclosed  material  system  is  maintained  at  a  temperature  T, 
the  interior  of  the  system  contains  energy  constantly  radiated 
to  and  from  the  boundary.  When  these  energy  exchanges  arrive 
at  equilibrium  each  cubic  centimentre  of  the  system  contains 
energy  in  what  we  may  call  the  undulatory  form.  The  problem 
is  how  to  calculate  the  mcst  probable  distribution  of  the  energy 
between  the  various  wave-lengths  not  only  for  the  single  tempera- 
ture T  but  for  any  temperature  ;  for  the  equilibrium  state  may 
be  defined  as  that  for  which  the  distribution  of  energy  (at  the 
given  temperature)  between  the  various  wave  lengths  is  the 
most  probable.  To  work  out  this  statistical  problem  we  must 
know  something  about  the  number  of  degrees  of  freedom  pos- 
sessed by  the  matter  and  by  the  ether  (present  throughout  the 
matter)  respectively.  It  will  now  be  shown  that  absolutely 
different  results  are  arrived  at,  according  as  to  whether  we  either 
regard  the  ether  as  continuous  (*".*.  structureless)  or,  on  the 

1  A  simple  but  at  the  same  time  comprehensive  account  of  the 
principles  upon  which  Planck's  theory  is  founded  is  contained  in  a  paper 
entitled  "An  elementary  account  of  the  quantum  theory,"  by  J.  Rice, 
Trans.  Faraday  Soc.,  1915,  Vol.  X.,  Part  2. 


476       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

other  hand,  regard  it  as  having  a  structure.  Let  us  first  consider 
the  ether  as  continuous.  In  this  case  the  ether  is  a  medium 
capable  of  vibrating  in  an  infinite  number  of  ways,  the  wave 
lengths  propagated  throughout  it  having  all  possible  values 
between  o  and  oo  .  This  is  the  same  thing  as  saying  that  the 
ether  possesses  an  infinite  number  of  degrees  of  freedom. 
Matter  imbedded  in  the  ether  has  on  the  other  hand  a  finite 
number  of  degrees  of  freedom  in  virtue  of  its  discontinuity, 
i.e.  in  virtue  of  its  discrete  or  heterogeneous  structure.  If  now 
we  apply  the  principle  of  equipartition  of  energy  to  such  a 
system  composed  of  both  matter  and  ether,  it  is  clear  that 
the  ether  will  take  all  the  energy  (since  its  number  of  freedoms 
is  infinite)  leaving  none  at  all  for  the  matter.  It  is  impossible 
therefore  to  conceive  of  an  equilibrium  being  set  up  as  regards 
energy  interchange  between  matter  and  ether  except  at  the 
absolute  zero  of  temperature.  In  other  words,  we  cannot 
deduce  any  radiation  law.  It  is  clear,  therefore,  we  must 
follow  out  some  other  line  of  reasoning.  Let  us  take  the 
second  case,  namely,  the  assumption  that  the  ether  does 
possess  a  structure.  On  this  basis  the  number  of  degrees 
of  freedom  of  the  ether  is  no  longer  infinite  and  it  is  possible 
to  conceive  of  equilibrium  states  being  reached  at  different 
temperatures,  as  a  result  of  energy  transfer  between  matter 
and  ether.  If  we  think  of  the  ether  as  possessing  a  fine- 
grained structure,  we  must  expect  that  the  waves  which  can  be 
propagated  by  such  a  medium  must  not  become  shorter  than 
a  certain  limiting  size  AQ.  It  is  easy  to  see  this  by  analogy. 
Waves  which  can  be  transmitted  by  a  material  system  (sound 
waves,  for  example)  must  be  great  compared  to  the  distance 
of  the  molecules  which  compose  the  system,  as  otherwise  the 
waves  would  not  be  transmitted  at  all.  Similarly  we  must 
suppose  that  even  the  shortest  light  waves  which  we  know  of 
must  be  large  compared  to  the  grain  structure  of  the  ether 
itself.  If  this  structure  exists  it  ceases  to  be  legitimate  to 
speak  of  infinitely  short  waves  in  the  mathematical  sense. 
The  shortest  conceivable  waves  must  be  of  the  order  of 
magnitude  of  the  distance  apart  of  the  "  molecules  of  the 
ether." 


THE   QUANTUM   THEORY  477 

Starting  out  with  the  idea  that  the  number  of  degrees  of 
freedom  possessed  by  the  ether  is  finite,  Jeans  has  shown  that 
by  applying  the  principle  ofequipartition  of  energy,  the  energy 
distributes  itself  in  the  normal  spectrum  in  such  a  way  that  the 
intensity — corresponding  to  a  region  lying  between  A  and 
A  +  d\ — is  proportional  to  the  temperature  and  inversely 
proportional  to  the  fourth  power  of  the  wave-length.  That 
is  to  say,  the  smaller  the  wave-length  (i.e.  the  greater  the 
vibration  frequency)  the  greater  is  the  number  of  degrees  of 
freedom  to  which  a  radiation  corresponds  in  any  given 
spectral  region.  In  other  words,  the  number  of  wave-lengths 
confined  between  A  and  d\  being  greater  the  smaller  the  value 
of  the  A,  the  degrees  of  freedom  corresponding  to  the  shortest 
wave-lengths  tend  to  take  up  all  the  energy  ;  that  is  the  energy 
in  the  spectrum  will  so  distribute  itself  that  it  will  be  almost 
entirely  confined  to  the  region  of  extremely  short  wave-lengths. 
As  a  matter  of  fact,  however,  this  is  not  the  real  distribution 
of  energy  in  the  spectrum.  It  has  been  shown  experimentally 
that  the  intensity  of  energy  radiating  from  a  black  body  and 
in  equilibrium  with  the  body,  shows  a  maximum  for  waves 
in  the  infra-red  region  at  ordinary  temperatures,  and  vanishes 
almost  completely  for  very  long  or  very  short  waves.  Repre- 
senting it  graphically  we  obtain  a  curve  similar  to  that  repre- 
senting the  Maxwell  distribution  of  velocities  of  molecules 
based  on  probability,  as  already  shown  in  the  preceding 
chapter  (Fig.  90). 

We  are  thus  met  once  more  with  a  serious  difficulty.  The 
assumption  that  the  number  of  degrees  of  freedom  of  the 
ether  is  not  infinite,  is  in  itself  insufficient  to  yield  an  ex- 
pression (involving  the  equipartition  principle)  which  will 
agree  with  experiment.  Planck  was  therefore  led  to  make  yet 
another  assumption.  According  to  this  assumption  the  ex- 
changes of  energy  between  ether  and  matter,  instead  of 
taking  place  in  any  proportion  whatsoever,  can  only  take  place 
by  steps,  that  is,  in  multiples  of  some  small  energy  unit.  In 
fact,  radiant  energy  itself  is  considered  to  possess  a  structure. 
At  the  present  time  (1915)  it  would  be  more  proper  to  say 
that  the  definite  concept  of  radiant  energy  in  itself  being 


478       A   SYSTEM  OF  PHYSICAL    CHEMISTRY 

discrete  in  nature  is  that  upheld  by  Einstein,  a  view  which 
has  been  given  a  clearer  physical  basis  by  J.  J.  Thomson, 
who  considers  a  radiant-energy  unit  or  "  quantum  "  as  a  region 
of  periodic  disturbance  travelling  along  a  Faraday  tube.  In 
place  of  a  continuous  ether  we  have  therefore  to  substitute  a 
number  of  stretched  strings  of  ether,  each  string  being  a 
Faraday  tube  differentiated  in  some  (practically  unknown)  way 
from  the  "  space"  surrounding  it.  This  statement,  is  however, 
to  be  taken  rather  as  a  rough  material  analogy  than  as  an 
exact  description  or  formulation,  for  any  exact  description  is 
at  the  present  time  impossible.  Planck,  on  the  other  hand, 
lays  the  greater  stress  not  on  the  question  of  the  ultimate 
structure  of  radiant  energy  itself  so  much  as  on  the  mode  of 
its  absorption  and  emission  by  matter.  The  energy  radiated 
by  one  element  of  a  black  body  is  partially  absorbed  by  other 
elements.  Each  one  of  the  vibrators  or  "  resonators,"  as 
Planck  calls  them,  which  constitute  the  material  element  in 
question,  can  only  emit  (and  absorb)  energy  in  certain  fractions. 
(This  at  any  rate  is  Planck's  first  position  ;  we  shall  see  later 
that  he  has  modified  the  above  view.)  A  word  here  about  the 
Planck  resonators.  It  was  pointed  out  in  the  preceding  chapter 
that  the  vibrating  particles  which  produced  visible  and  ultra- 
violet light  were  much  smaller  than  the  atoms  of  the  substance 
and  were  therefore  in  all  probability  the  electrons.  The  amount 
of  energy  contributed  to  the  "total"  spectrum  by  the  visible  or 
ultra-violet  region  has  been  shown  by  experiment  (cf.  Lummer 
and  Pringsheim's  curve,  Fig.  90)  to  be  a  very  small  fraction  of 
the  whole  (even  at  fairly  high  temperatures).  The  main  part  of 
the  energy  is  confined  to  the  infra-red  region.  These  long  waves 
are  considered  as  emitted  and  absorbed  by  the  atoms  of  the  sub- 
stance being  set  in  vibration.  These  statements  apply  only  to 
the  continuous  spectrum  given  out  by  a  black  body  in  the  first 
instance  and  in  general  by  heated  metals.  Planck's  resonators 
for  the  infra-red  may  therefore  be  identified  as  the  atoms 
(electrically  charged),  for  the  ultra-violet  the  electrons.  But 
Planck  in  the  theoretical  treatment  of  the  subject  has  regarded 
his  resonators  as  linear^  that  is,  he  only  considers  the  energy 
caused  by  vibration  in  one  plane.  Such  a  vibration  entails 


THE   QUANTUM   THEORY  479 

one  degree  of  freedom.  If  we  were  to  apply  the  equipartition 
principle  to  a  system  of  such  linear  resonators,  to  each 
resonator  we  would  ascribe  RT  units  of  energy  made  up  of 
JRT  kinetic  energy  and  an  eq  livalent  quantity  of  potential. 
It  is  only  for  extremely  long  waves  that  the  equipartition 
principle  holds  (as  is  shown  by  the  degree  of  applicability  of 
Lord  Rayleigh's  formula  for  radiation)  and  that  only  for  a 
limited  temperature  range.  The  essence  of  PlancKs  view  is 
that  it  discards  the  equipartition  principle.  It  will  be  noted 
that  the  linear  resonators  of  Planck  are  considered  to  possess 
only  one-third  of  the  total  vibrational  energy  of  the  actual 
atoms  (of  a  solid),  each  of  which  possesses  three  degrees  of 
freedom — as  already  pointed  out  in  connection  with  the  values 
for  atomic  heat. 

According  to  Planck,  the  material  resonators  considered 
do  not  react  with,  or  are  not  influenced  by,  infinitely  small 
quantities  of  radiation  energy,  using  the  word  infinitely  in 
its  strict  sense.  Planck's  hypothesis  may  be  stated  thus :  It 
is  necessary  that  the  energy  exceeds  a  finite  value  e  in  order  that 
the  resonators  composing  the  material  system  may  be  able  to 
absorb  it  or  emit  it.  *  (As  already  mentioned,  Planck  later 
modified  this  statement  by  supposing  absorption  to  be  con- 
tinuous, but  emission  discontinuous,  i.e.  in  quanta.)  The  above 
statement  is  known  as  Planck's  "  unitary  theory  of  (radiant) 
energy."  It  will  be  clear  how  this  hypothesis  modifies  the 
principle  of  equipartition  of  energy  among  various  degrees  of 
freedom.  To  any  degree  of  freedom  which  actually  possesses 
energy  we  cannot  ascribe  less  than  one  quantum  (e),  and  the 
actual  quantity  possessed  will  be  an  integral  multiple  of  one 
quantum.  With  this  distribution  of  energy  some  of  the  degrees 
of  freedom  may  possess  no  energy  at  all,  i.e.  the  ^//-partition 
idea  breaks  down. 

The  term  "  the  quantum"  requires  now  to  be  considered. 
Although  we  speak  of  this  as  the  unit  of  (radiant)  energy, 
it  must  be  clearly  understood  that  //  is  NOT  a  fixed  and  constant 
quantity  of  energy.  According  to  Planck,  the  quantum  €,  i.e. 
its  size  or  magnitude,  is  a  function  of  the  vibration  frequency 
(either  of  the  radiation,  supposing  this  to  be  monochromatic, 


48o       A   SYSTEM  OF  PHYSICAL    CHEMISTRY 

or  what  amounts  to  the  same  thing,  the  frequency  of  the 
resonator).  In  fact,  According  to  Planck,  there  is  direct  pro- 
portionality between  the  magnitude  of  e  and  the  frequency  vt 
this  proportionality  being  expressed  in  Planck's  fundamental 
relation — 

€  =  hv 

where  h  is  a  universal  constant  (Planck's  constant)  having 
the  numerical  value  7  X  io~27  erg-seconds.1 

Now  the  smaller  the  unit  the  greater  the  probability  that  a 
resonator  will  possess  at  least  one  or  some  quanta.  If  we 
consider  a  material  system,  made  up  of  molecules,  atoms,  and 
electrons,  such  a  system  possesses  resonators  of  various  dimen- 
sions, i.e.  capable  of  vibrating  with  different  frequencies.  Such 
a  system  can  absorb  or  emit  a  range  (or  spectrum)  of  vibration 
frequencies.  Considering  the  very  short  waves,  i.e.  large 
vibration  frequency,  the  quantum  e  corresponding  to  this  is 
large,  and  hence  the  chance  that  a  resonator  possesses  even 
one  quantum  of  this  size  is  less  than  in  the  case  of  longer 
waves,  where  each  quantum  is  a  smaller  magnitude.  Less 
energy  of  the  short  wave  type  will  therefore  be  emitted  than 
that  of  the  longer  wave  type.  That  is,  the  energy  of  the 
radiation  emission  curve  falls  off  in  the  short  wave  region.  In 
this  way  Planck  explains  the  observed  diminution  in  energy 
emitted  in  the  visible  and  ultra-violet  region,  as  shown  in 
Lummer  and  Pringsheim's  curves.  Further,  in  the  region  of 
extremely  long  waves  v  is  relatively  very  small,  and  hence  the 
size  of  the  unit  €  is  small,  so  that  for  extremely  long  waves 
the  actual  energy  contribution  made  by  this  region  will  be 
small.  We  should,  therefore,  expect  on  Planck's  view  the 
energy  wave-length  curve  to  pass  through  a  maximum,  as  is 
actually  the  case. 

Starting  out  with  Planck's  hypothesis  of  the  discrete  nature 
of  radiation,  it  is  now  necessary  to  see  what  radiation  formula 
may  be  deduced ;  in  other  words,  what  theoretical  expression 

1  Planck  (Annalen  der  Physik,  [4]  4,  553,  1901)  has  shown  that  the 
magnitude  of  e  is  a  function  of  v  by  applying  Wien's  displacement  law  to 
an  expression  obtained  by  him  for  the  entropy  of  a  system.  The  reader 
should  also  consult  Planck's  Theorie  der  Wdrmestrahlung,  2nd  edition. 


THE   QUANTUM   THEORY  481 

can  be  deduced  for  the  distribution  of  energy  in  the  spectrum 
of  a  body  emitting  "  temperature,"  i.e.  black-body-radiation. 
For  an  exact  and  complete  account,  the  reader  is  referred  to 
Planck's  Theorie  der  Wdrmestrahlung.  We  can  only  here 
attempt  an  abbreviated  and  approximate  deduction,  based  upon 
a  new  method  employed  by  Jeans  (Phil.  Mag^  20,  953,  1910). 
If  a  vibration  —  that  is,  a  very  small  spectral  region  lying 
between  A  and  A  +  d\  which  corresponds  experimentally  to 
monochromatic  radiation  —  can  possess  the  following  amounts 
of  energy,  viz.  o,  e,  2€  .  .  .  etc.,  then  the  ratio  of  the  proba- 
bilities of  these  events,  as  in  the  usual  gas  theory  calcula- 
tions, is-  i  .  g_flKK  .  ^^T  .  etc< 

where  e  is  the  base  of  natural  logarithms,  k  a  constant, 
namely,  the  gas  constant  per  molecule,  and  T  the  absolute 
temperature.  This  means  that  if  we  represent  by  "  i  " 
the  number  of  vibrations  possessing  no  energy  at  all,  then 
the  number  of  vibrations  which  possess  e  units  each  will 
be  e~€/kT,  and  so  on.  Instead  of  thinking  of  vibrations  in 
"space,"  let  us  think  of  the  resonators  or  vibrations  of 
matter  which  can  emit  or  absorb  such  vibrations.  Let  us 
suppose  that  the  material  system  consists  of  a  great  number 
of  similar  vibrators  (emitting  and  absorbing  the  wave-length  A) 
each  acting  independently,  then  if  we  represent  by  unity  the 
number  of  such  resonators  which  possess  no  energy,  the  term 
c  -e/fcT  gives  the  number  of  resonators  possessing  e  units  each, 
etc.  If  out  of  N  such  resonators  under  consideration  M  have 
zero  energy,  the  number  which  have  each  energy  e  is  M>-e/feT, 
the  number  having  energy  2€  is  Md~2e/*T,  and  so  on.  Hence, 
for  the  total  — 

N  =  M  +  M*?-*/*1'  -f  M£-*/*T  -f  M*-3e/*T  -f  etc. 
=  M(i  +  e-*l™  +  e-*/kT  +  etc.) 
_  _  M 

-(l_^e/*T) 

And  if  27U  is  the  total  energy  of  the  N  resonators— 

MXo      eX  Mr-/*1  +  26  X  M*-2e/*T  +  ,  etc. 


T.P.C.  —  II. 


482       A   SYSTEM  OF  PHYSICAL    CHEMISTRY 
Substituting  the  value  for  N  already  given,  we  obtain- 


If  we  now  substitute  hv  for  c  we  obtain  — 


And  for  the  energy  of  one  such  resonator  we  obtain  the  mean 
value — 

U  -         hv 

u  — 


So  much  for  the  energy  of  a  single  vibrating  resonator  in 
radiation  equilibrium  with  its  surroundings,  the  expression 
being  based  on  the  hypothesis  that  energy  is  made  up  of 
units,  the  magnitude  of  a  unit  being  directly  proportional  to 
the  vibration  frequency  of  the  resonator.  We  have  now  to 
see  the  connection  between  this  term  U  and  the  radiation 
density  uv,  that  is,  the  quantity  of  monochromatic  radiation 
(frequency  v)  energy  per  c.c.  of  radiated  space.  [In  actual 
practice  a  heated  body  does  not  give  out  monochromatic 
radiation,  but  a  complete  spectrum.  We  shall  define  uv  in 
different  terms  later,  though  still  equivalent  to  the  above.] 

Planck  has  shown  on  the  basis  of  the  classical  electro- 
magnetic theory  (which  therefore  introduces  no  unitary  hypo- 
thesis), that  uv  and  U  are  connected  by  the  relation  — 


[Cf.  Planck,  Annalen  der  Physik.^  [4]  4,  560,  1901  ;  Linde- 
mann,  Brit.  Ass.  Rep.>  1912.  If  we  take  A—  8oo/z/i  as  a 
mean  value  for  a  wave-length  in  the  red  part  of  the  spectrum, 
it  is  easily  calculated  from  this  formula  that  U  =  Su  (approx.). 
Similarly  for  the  violet  end  of  the  spectrum,  taking  A  =  400^, 
one  finds  U  =  2U  (approx.).] 

By  combining  the  two  expressions  obtained  above,  the 
mean  value  of  the  density  of  energy  radiated  from  a  single 
resonator  in  a  system  consisting  of  a  large  number  of  similar 


PLANCICS  RADIATION  FORMULA  483 

resonators,  all  emitting  monochromatic  light  of  frequency  vt  is 
given  by  the  expression — 


_ 

Uv  ~ 


This  is  one  of  the  forms  of  Planck's  Radiation  Formula. 

We  wish  now  to  change  the  shape  of  this  expression  a 
little,  in  order  to  be  clear  about  the  relation  of  the  term  uv  and 
the  term  EA,  which  latter  we  have  already  met  with  in  Wien's 
radiation  formula,  for  example. 

The  total  energy  (say  in  ergs)  radiated  per  second  from 
unit  area  of  a  black  body  emitting  a  continuous  spectrum 
covering  the  wave-lengths  o  to  co  has  been  denoted  by  S. 
This  is  the  term  which  appears  in  Stefan's  Law,  viz.  S  =  o-T4, 
the  temperature  of  the  source  being  T  and  the  radiation 
purely  "  temperature  "  radiation.  Now  S  is  the  quantity  of 
energy  which  would  be  present  in  an  imaginary  cylinder,  i 
square  centimetre  base  and  length  3  X  ioi°  cms.  (since 
3  X  io10  cms.  is  the  distance  which  the  radiation  will  travel 
in  one  second).  The  cylinder  is  supposed  to  be  placed  with 
its  base  on  the  radiating  body  and  extending  out  into 
space.  The  volume  of  this  cylinder  is  3  X  io10  c.c.,  and 

since  this  contains  S  ergs  of  energy  the  space  density  of  the 

g 
radiation,  i.e.  the  amount  of  energy,  is  10  erg  per  c.c. 

Denoting  the   space  density  of  the   energy  by  E,  we  have 

c 

£  _  -  —  __  or   g  —  3  x  io10E,   the    term  E  referring,  of 

course,  to  the  entire  range  of  wave-lengths  between  o  and  oo 
emitted  by  the  body.  This  total  density  E  may  be  expressed 
thus  — 


</E  /"°° 

Let  us  write  Ex  =  ^,  then  E  =J 


It  will  thus  be  seen  that  EA  is  a  rale  (the  rate  of  change  of 
E  with  A).     We  might   also  define  it  as   the  density  of  the 


484       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

energy  radiated  from  a  spectral  region  the  wave-length  limits 
of  which  differ  by  tmity.  This,  though  correct,  is  physically 
inconceivable.  It  would  entail  the  existence  of  a  spectrum 
the  wave-length  limits  of  which  differ  by  i  cm.,  a  difference 
enormous  compared  to  any  actual  limits  reached.  It  is  there- 
fore much  less  confusing  to  think  of  EA  as  being  the  small 
energy  density  increment  AE  divided  by  the  correspondingly 

•at? 

small  spectral  width  AA  (or  more  accurately  -^-  as  above). 

The  expression  EA*/A  or  •**•  dX  is  therefore  the  energy-density 

of  the  radiation  between  A  and  A  -f  dX.  Planck  also  gives  the 
same  significance  to  the  term  uvdv.  Thus  the  energy-density 
of  the  total  spectrum  may  be  written — 


or  writing  uv  =  .-  ,  E  =  I    uvdv 

The  term  uv  is  also  a  rate  or  it  may  be  denned  as  the  density 
of  the  energy  radiated  from  a  spectral  region,  the  limits  of  the 
vibration  frequency  differing  by  unity.  Numerically  uv  is  of 
quite  a  different  order  of  magnitude  from  EA.  Thus,  since 

A  =  -,  where   c  is   the  velocity   of  light,  it   is   evident   that 

—  = ,  so  that  on  increasing  v  by  unity  (i.e.  corresponding 

to  the  production  of  uv  energy-density  units)  the  wave-length 
decreases  by  the  amount  2 .  This  gives  a  result  of  the  order 

IO-i8  in  the  case  in  which  the  infra-red  wave-length  region 
A  =  iju,  is  considered.  Since  EA  is  the  energy-density  corre- 
sponding to  a  wave-length  difference  of  unity r,  the  term  EA 
is  io18  uv  for  the  region  in  which  the  vibration  frequency 
differs  by  unity. 

In  bolometric  or  spectro-photometric  determinations  one 
measures  the  energy  AS  radiated  per  second  by  a  small  region 


PLANCK'S   RADIATION  FORMULA  485 

of  a  continuous  spectrum  which  lies  between  A  and  A  -f-  AA. 
Since  S  =  3  X  ioi°  E,  and  therefore  AS  =  3  X  ioi°  AE,  we 

AE 

can  calculate  ^r  which  in  turn  is  identical  with  Ev     We  now 

wish  to  express  the  radiation  formula  of  Planck  in  such  a  way 
as  to  allow  of  convenient  comparison  between  EA  observed 
and  its  value  as  given  by  the  formula.  To  do  this  it  is 
necessary  to  write  the  formula  in  the  shape  — 


And     remembering    that     uvdv  =  E^A,    and     further     that 
;/A—  —  -dV,  we  obtain  directly  — 


which  is  the  more  usual  form  of  Planck's  equation  for  the 
distribution  of  energy  throughout  the  spectrum.  As  already 
pointed  out,  we  can  compare  the  observed  and  calculated 
values  of  EA  for  a  whole  series  of  small  regions  and  thus  test 
the  equation  over  the  complete  spectrum.  It  has  been  found 
that  the  above  formula  reproduces  the  actual  energy  distribu- 
tion from  a  black  body  in  an  extremely  accurate  manner, 
being  in  fact  the  most  satisfactory  equation  yet  proposed. 
The  principal  experimental  investigations  are  those  of  Lummer 
and  Pringsheim  (Verh.  d.  Dents  ch.  physik.  Gesell.^  1,  23  and 
215,  1899;  2,  163,  1900),  Rubens  and  Kurlbaum  (Sitzungsber. 
d.  K.  Akad.  d.  Wissenschaft  zu  Berlin^  41,  929,  1900  ;  Annalen 
d.  Physik)  [4]  4,  649,  1901),  Paschen  (Annalen  d.  Physik^  [4] 
4,  277,  1901)  and  the  more  recent  measurements  of  W.  Cob- 
lentz  (Bull.  Bur.  Standards,  1914,  10,  p.  i). 

By  way  of  illustration  a  few  of  Paschen  's  results  may  be 
quoted.  The  extent  of  the  spectrum  examined  was  from  i  to 
8'8/x  (i.e.  the  infra-red  region),  measurements  being  carried 
out  by  means  of  the  bolometer.  Using  a  fluorite  prism  (which 
is  more  transparent  than  quartz)  successive  parts  of  the 
spectrum  could  be  isolated.  The  radiating  body  consisted 
of  a  hollow  vessel  or  cylinder,  electrically  heated  to  any 


486       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

desired  temperature,  the  cylinder  being  either  of  platinum  or 
porcelain  with  or  witTiout  a  coating  of  copper  oxide.  In  the 
following  table  /  denotes  the  magnitude  of  the  swing  of  the 
galvanometer  needle,  a  certain  swing  corresponding  to  a 
certain  amount  of  energy  communicated  by  the  radiation  to 
the  bolometer.  After  calibrating  the  apparatus,  Planck's 
formula  can  be  used  to  calculate  values  of  i  for  different  wave 
regions,  and  these  may  be  compared  with  experiment. 


TABLE  I. — A  =  1*0959  /*.    Width  of  spectrum  isolated  =  3'. 
Temperature  of  surroundings  9-8°  C. 


T  =  temperature  of  radiator  = 

i333'4 

1553*1 

1628-3 

1038-7 

(Observed     

60'6 

2C2-6 

CO'? 

3-65 

il  Calculated  from  Wien's  equation 
(Calculated  from  Planck's  equation 

6  1  "44 
61-27 

252^ 
249-4 

59'H 
58'97 

y6\ 

3-68 

TABLE  II. — A  =  8-7958  /z.     Width  of  spectrum  isolated  =  6'. 
Temperature  of  surroundings  =  15*1°  C. 


| 

emperature  of  radiator  = 

1458-6           1069-8 

844-4 

625*6 

483*1 

Observed    . 

131-7 

74*43 

44'  5  i 

20-46 

9-22 

Wien's  equation    . 

90-40          59-79 

39-52         19-86 

9-08 

Planck's  equation  . 

129-1 

7375 

44'  83 

20-90 

9-22 

It  will  be  observed  that  Planck's  formula  is  much  more 
applicable  than  Wien's  for  high  temperatures  and  long  waves. 
Besides  the  infra-red  region,  the  experiments  of  Paschen  and 
Wanner,  as  well  as  those  of  Lummer  and  Pringsheim,  on 
radiation  from  the  visible  spectral  region,  have  shown  that 
Planck's  formula  holds  here  also  within  the  limits  of  experi- 
mental error.  (We  shall  have  occasion  later  on  to  return  to 
Rubens'  and  Kurlbaum's  "  Reststrahlen "  method  of  investi- 
gating regions  very  far  in  the  infra-red.)  In  addition  to  the 
direct  method  of  testing  Planck's  formula  outlined  above,  its 
validity  is  further  shown  by  the  fact  that  we  can  deduce  from 


PLANCK'S  RADIATION  FORMULA  487 

it  directly  Wien's  displacement  law  (viz.  A^ax.  X  T  =  constant, 
and  Emax.  =  constant  X  T5),  and  Stefan's  total  radiation  law 

=  const.  X  T*  where  S  =  3  X  ioi°  E  =  3  X  TO™  f"  KA 

both  of  which  have  been  experimentally  verified.     Planck's 
expression  is  thus  a  very  comprehensive  one.1 


Some  Numerical  Values. 

At  this  point  it  is  of  interest  to  calculate  the  values  of  the 
two  fundamental  constants  h  and  k  which  occur  in  Planck's 
equation.  //  is  the  universal  proportionality-factor  connecting 
the  energy  e  of  a  quantum  with  the  frequency  v ;  £,  as  will  be 
shown  later,  is  the  gas  constant  R,  reckoned  not  for  a  gram- 
mole  but  for  a  single  molecule.  The  calculation  may  be  carried 
out  as  follows. 

Kurlbaum  (Wied.  Ann.  65,  759,  1898)  has  found  by  experi- 
ment that  the  total  energy  emitted  from  i  square  centimetre 
of  a  "  black  body  "  in  i  second,  the  temperature  of  the  body 

1  To  emphasise  further  the  fact  that  Planck's  formula  is  at  variance 
with  the  principle  of  ^/-partition  of  energy  among  various  degrees  of 
freedom,  it  is  interesting  to  calculate  the  energy  of  an  electron  vibrating 
with  a  frequency  identical  with  that  of  the  ultra-violet  region  when  light  is 
emitted,  and  compare  this  energy  with  the  energy  of  an  atom  or  of  a 
gaseous  molecule  possessing  the  mean  kinetic  energy  characteristic  of 
ordinary  temperature  (say  300°  absolute)  (cf.  J.  Stark,  Zeitsch.  physik. 
Chem.,  86,  53,  1913).  The  frequency  v  for  the  ultra-violet  region  will  be 
about  5  x  iou  per  second.  Planck's  expression  for  the  mean  energy  of  a 
single  resonator  is,  as  we  have  already  seen  — 

hv 

U  =  >'**  -  i 
For  ordinary  (low)  temperatures — T  =  300° — this  expression  reduces  to 

_hv 

U  =  hve  ~  *T ,  since  v  is  large  and  T  small.  The  mean  energy  of  an 
electron  resonator  is  therefore  U,  where  U  =  i  X  io~16  ergs.  The  mean 
kinetic  energy  of  a  gas  molecule  at  the  ordinary  temperature  =  3/2#T 
=  6  x  IQ-14  ergs,  so  that  instead  of  equipartition  of  energy  we  find  that 
the  energy  of  the  electron  vibrating  in  the  atom  or  molecule  is  only  one 
fivehundredth  part  of  the  mean  kinetic  energy  of  translation  of  the  molecule 
itself. 


438        A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

being  100°  C.  and  that  of  the  air  being  taken  as  o°  C,  amounts 
to  0*0731  watt/cm2,  that  is  — 

S  =  Sioo>  c.  —  So°  c.  =  4'  2  X  7*31  X  io5  ergs/cm.2-sec. 
By  applying  Stefan's  Law  we  obtain  — 


Hence       ,  - 

(373—273')      cm.2-sec.-degrees4 

The  physical  significance  of  cr  is  evidently  the  total  radiation 
emitted  from  a  black  body  per  second,  when  the  temperature 
difference  between  the  black  body  and  the  surroundings  is  i 
degree,  the  temperature  of  the  black  body  being  i°  absolute, 
the  surroundings  being  at  o°  absolute.  Corresponding  to  this 
emission  at  i°  absolute,  we  see  that  E,  i.e.  the  total  energy 

density,  is  —j,  that  is— 


Now  E  =  /   Ex</A,  or    /    uvdv  and  Ex  or  uv  is  given  by 
J  o  '  o 

Planck's  expression.     Using  Planck's  expression  in  the  form 
in  which  uv  occurs,  we  get  — 


8** 


the  term  T  being  omitted  in  the  final  expression,  since  it  is 
simply  unity.  The  integration  may  be  effected  by  series,  and 
we  obtain  finally  — 

4877V&4 


Setting  this  equal  to  the  "  observed"  value  of  Ei°abs.,  viz. 
7*061  X  io~15  we  obtain  — 

^=  I'l682  X   1016    .....       (3) 

Further,  Lummer  and  Pringsheim  (Verh.  d.  Deutsh.  physik. 


PLANCK^S  RADIATION  FORMULA  489 

Gesell.,  2,  176,  1900)  have  determined  the  value  of  Amax.T, 
where  Amax.  is  the  wave-length  corresponding  to  the  maximum 
value  of  EA  from  a  black  body  radiating  at  a  given  temperature. 
The  expression  Amax.  X  T  is  a  constant  independent  of  tem- 
perature as  Wien  has  shown,  the  numerical  value  found  by 
Lummer  and  Pringsheim  being  0*294  cm.  degrees.  Now,  by 
differentiating  Planck's  formula  (equation  (2)),  with  respect  to 
A,  and  putting  the  differential  equal  to  zero,  when  A  =  Amax. 
we  obtain — 

(,_    <*> 

5>eAmax.  1  / 

whence  Amax.  X  T  =    .  * 

or  T  = 


By  combining  equations  (3)  and  (4)  we  obtain  finally— 

h  —  6-55  X  iQ-27  erg/sec. 
k  =  1*346  X  io~16  erg/degree 

A   more  recent  recalculation    of  h  has  raised  the  value 
somewhat,  viz.  to  7  X  io~27  erg./sec. 


The  Significance  of  the  constant  k,  and  a  Determination  of  the 
number  of  Molecules  in  one  gram-molecule. 

Consider  once  more  the  Planck  expression  for  the  energy 
of  vibration  of  a  large  number  of  similar  resonating  particles 
emitting  monochromatic  radiation,  viz.  — 

27U-       Nc 

- 


The  expression  <?6/fcT  —  i  may  be  expanded  thus  — 


/r  +  I&T  )2  +  higher  powers 

If  now  we  are  dealing  with  a  system  vibrating  at  very  high 


490       A   SYSTEM   OF  PHYSICAL   CHEMISTRY 
temperature  it  will  be  seen  that  the  above  expression  becomes 
j     .     That  is,  at  high  temperatures  — 


Exactly  the  same  result  is  obtained  at  less  high  tempera- 
tures if  the  system  is  vibrating  very  slowly,  for  in  this  case  v 
is  small  (relatively),  and  since  €  =  hv  the  quantity  t  is  like- 
wise small.  In  both  cases  the  quantity  e  vanishes  from  the 
expression  for  the  sum  of  the  energies  of  the  vibrating  particles, 
this  energy  being  simply  proportional  to  the  absolute  tem- 
perature. Under  these  conditions  we  reobtain  the  results  of 
the  ordinary  kinetic  theory,  i.e.  the  principle  of  equipartition 
of  energy,  it  being  no  longer  necessary  to  consider  the  energy 
as  other  than  continuous.  The  principle  of  equipartition  of 
energy  is  therefore  true  as  a  limiting  case  for  large  values  of 
T,  and  for  small  values  of  v.  Suppose  that  we  are  dealing 
with  a  solid  radiating  energy  at  a  temperature  sufficiently  high 
that  the  energy  of  vibration  of  the  resonators  could  be  repre- 
sented by  N^T.  The  resonators,  as  employed  by  Planck,  are 
linear,  i.e.  they  possess  i  degree  of  freedom,  to  which  one 
would  ascribe  (if  the  equi-partition  principle  applied,  i.e.  if  T 
is  sufficiently  high)  JRT  kinetic,  and  JRT  potential,  in  all 
RT  units  of  energy  per  gram-mole  or  gram-atom,  if  a  mon- 
atomic  solid  be  considered,  and  if  the  atoms  be  identified  with 
the  resonators. 

Hence,  the  term  J£U  reckoned  for  N  atoms  where  N  is 
now  regarded  as  the  number  of  atoms  in  a  gram-atom 
(or  molecules  in  a  gram-mole),  should  be  identical  with  RT. 
That  is,  RT  =  £NT,  or  R  =  £N.  The  constant  k  is  therefore 
simply  the  gas  constant  R  reckoned  for  a  single  molecule. 

We  may  see  in  another  way  that  k  has  this  significance. 
As  has  been  pointed  out  already,  Jeans  l  has  shown  that  for 
very  long  waves  the  law  of  the  partition  of  energy,  or  of 
energy-density  between  waves  of  different  lengths,  as  expressed 

1  Jeans,  Phil.  Mag.,  [6],  17,  229,  1909. 


PLANCK'S  RADIATION  FORMULA  491 

in  a  formula,  must  contain  the  wave-length  term  to  the  inverse 
fourth  power.  A  similar  formula  was  deduced  by  Lord 
Rayleigh  on  the  classical  theory,  and  holds  well  for  the  very 
long  wave-length  region.  Jeans'  equation  is  — 

EA  =  87TR/TA-4 

where  R?  is  here  the  gas  constant  reckoned  for  a  single  molecule. 
Now  for  this  very  long  wave  region  we  have  seen  that  Planck's 

expression  may  be  simplified,  the   term  -^*  ------  becoming 


—  or  -=—  . 
hv       he 

That  is,  Planck's  equation  becomes  — 
Zithc 


On  comparing  this  with  Jeans'  equation,  it  is  at  once 
evident  that  the  two  become  identical  —  and  they  must  be 
identical  if  they  are  equally  to  reproduce  experimental  results 
in  the  long-wave  region  —  if  k  is  identical  with  Rj,  t.e.  k  is  the 
gas  constant  per  single  molecule.  This  may  be  tested  at  once 
by  calculating  N  from  the  known  value  R  (per  mole),  and  the 
value  of  k  calculated  from  radiation  data,  using  Planck's  formula. 
In  this  way  it  is  found  that  N  =  6*175  X  io23,  a  number  which 
agrees  very  well  indeed  with  the  values  obtained  by  Perrin 
and  Miliikan  (cf.  Vol.  I.  Chap.  I.).  This  "  radiation"  method 
may  therefore  be  regarded  as  a  new  and  independent  method 
of  calculating  the  Avogadro  constant.  The  value  of  N  just 
given  leads  to  a  value  for  the  charge  on  an  electron,  viz. 
4*69  X  io~10  electrostatic  units,  which  agrees  well  with  Mil- 
likan's  value,  4777  X  io-10. 


Numerical  Values  of  the  Size  of  the  quantum  €  in  different 
Spectral  Regions. 

In  the  following  table  are  collected  some  values  of  (/iv) 
extending  over  a  wide  range.  The  longest  wave  measured  by 
Rubens,  in  1910,  was  A  =  90^,  though  still  longer  waves  have 


492       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

been   measured    more    recently.      The   shortest   wave-length 
measured  is  of  the  order  o'l/x,  or  xoo/x/x  (Schumann). 


Wave-length  A. 

c 

r=Br 

€  =  hv. 

A  ~  oou 

•2"}    X    IO12 

2*  I  x  io~14    ergs 

A  -  6/z  .          

5v   IO13 

1"i  x  ID"13 

A  =  2/1.     (This  is  the  region  of 
maximum  intensity  when   the 
body  is  at  1646°  absolute) 

i'S  x  io14 

9-9  x  io-13      „ 

A  =  o'Sji.   (This  is  approximately  } 
the  limit  of  the  visible  red)       / 

375  x  IOM 

2-47  x  icr  12    „ 

A  =  O'4/i.   (This  is  approximately  } 
the  limit  of  the  visible  violet)   / 

7-5  x  io14 

4-95  x  io-12    „ 

A  =  O'2/x.     (This  is  the  limit  ob-1 
tainable  with  a  quartz  prism)    / 

i  '5  x  io15 

9-9  x  io~12      ,, 

A  =  o'ljM.     Schumann  rays    .      . 

3-0  x  io15 

1-9  x  io-11      „ 

The  value  of  €  can  thus  vary  from  io-11  to  io~14  ergs, 
according  to  the  vibration  frequency  of  the  resonator,  i.e.  the 
vibration  frequency  of  the  radiation.  As  already  pointed  out, 
the  resonators  emitting  short  waves  are  probably  electrons, 
those  emitting  long  waves  are  probably  the  atoms.  As  regards 
the  total  energy  emitted  from  a  heated  body,  we  can  neglect 
the  portion  due  to  wave-lengths  shorter  than  o'Sjji,  i.e.  we  can 
neglect  the  visible  and  ultra-violet  regions.  This  is  true,  of 
course,  only  for  bodies  the  temperature  of  which  is  not  higher 
than  1600°  C.  If  the  temperature  of  the  radiator  be  raised 
much  higher  than  this,  the  contribution  from  the  visible  may 
no  longer  be  negligible,  since  in  accordance  with  Wien's  dis- 
placement law  the  position  of  maximum  energy  emission  shifts 
towards  the  shorter  wave  region  the  higher  the  temperature. 
A  body  with  the  temperature  of  the  sun  (say  6000°  C.),  radiat- 
ing purely  thermally  (without  chemical  effects  in  addition, 
as  is  the  case  with  the  sun),  would  have  its  maximum  energy 


THE   PHOTO-ELECTRIC  EFFECT  493 

emission  at  A  =  o'5/z  approximately,  i.e.  the  green-blue  region. 
Experimental  realisation  of  such  a  temperature  is  quite  im- 
possible. 


EXPERIMENTAL  EVIDENCE  IN  FAVOUR  OF  THE  DISCRETE 
NATURE  OF  RADIANT  ENERGY. 

The  fact  that  Planck's  formula  reproduces  experimental 
values  over  the  entire  range  of  the  spectrum  investigated  is, 
of  course,  strong  evidence  in  favour  of  the  validity  of  the 
assumptions  upon  which  the  formula  is  based,  i.e.  the  discrete 
nature  of  radiation.  Further,  as  will  be  shown  later,  the 
evidence  from  the  standpoint  of  the  specific  heat  of  solids 
and  its  variation  with  temperature  is  of  even  a  more  con- 
vincing nature.  In  addition  to  this,  there  is  evidence  of 
perhaps  a  more  direct  nature,  obtainable  from  the  results  of 
investigation  of  the  phenomenon  known  as  the  photo-electric 
effect.  Experiment  has  shown  that  when  metals  in  general, 
and  zinc  in  particular,  are  exposed  to  light  of  suitable  wave- 
length they  emit  negatively  charged  particles  (electrons). 
Using  ultra-violet  light,  the  photo-electric  effect  is  observed, 
not  only  in  the  case  of  metals  (especially  the  alkali  metals), 
but  even  in  the  case  of  non-metallic  substances,  e.g.  water, 
aqueous  solutions,  salts,  and  certain  crystals.  It  has  been 
shown  that  in  all  cases  this  emission  of  negative  electricity  is 
bound  up  with  an  absorption  of  the  light.  The  periodic 
electric  force  in  the  radiation  sets  the  electrons  in  more 
violent  vibration,  so  that  a  number  are  projected  from 
the  metal  into  the  neighbouring  space.  It  is  evident  that 
the  photo-electric  current  produced  by  the  moving  electrons 
depends  on  two  factors,  (i)  the  number  of  electrons  emitted 
in  unit  time,  and  (2)  the  speed  with  which  these  electrons  are 
emitted.  If  the  intensity  of  the  light  be  increased  without 
altering  the  colour,  i.e.  without  altering  the  vibration  frequency, 
it  has  been  found  that  the  photo-electric  effect  increases  pro- 
portionally to  this  intensity,  because  the  number  of  electrons 
emitted  increases  in  the  same  proportion.  On  the  other  hand, 
working  with  constant  intensity  but  varying  the  colour  in 


494       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

the  sense  of  increasing  the  frequency  of  the  light,  the  photo- 
electric effect  has  likewise  been  observed  to  increase.  The 
experiments  of  Lenard  and  Ladenburg  appear  to  indicate  that 
when  the  wave-length  is  shortened  (at  constant  intensity),  the 
speed  of  emission  of  the  electrons  increases  proportionally  to 
the  frequency.  If  the  speed  of  the  electrons  is  due  to  the 
light,  how  is  it  that  it  is  the  same  for  a  very  feeble  and  for 
very  intense  light  (of  the  same  wave-length  in  each  case),  and 
still  more,  how  are  we  to  account  for  the  variation  of  the 
speed  of  the  electrons  as  proportional  to  the  frequency  (the 
intensity  being  kept  constant)  ?  Let  us  see  what  explanation 
is  offered  on  the  assumption  that  light  is  "  heterogeneous  "  in 
structure.  One  cannot  do  better  than  quote  Sir  J.  J.  Thomson's 
words  (Proc.  Camb.  Phil.  Soc.,  14,  421,  1908) : — "  The  [radiant] 
energy  travelling  outwards  [from  the  radiating  source]  with  the 
wave  is  not  spread  uniformly  over  the  wave  front,  but  is  con- 
centrated on  those  parts  of  the  front  where  the  pulses  are 
travelling  along  the  lines  of  force ; 1  these  parts  correspond 
to  the  bright  specks,  the  rest  to  the  dark  ground.  .  .  .  The 
energy  of  the  wave  is  thus  collected  into  isolated  regions, 
these  regions  being  the  portions  of  the  lines  of  force  occupied 
by  the  pulses  or  wave  motion.  In  fact,  from  this  point  of 
view,  the  distribution  of  energy  is  very  like  that  contemplated 
on  the  old  emission  theory,  according  to  which  the  energy 
was  located  on  moving  particles,  sparsely  disseminated  through- 
out space.  The  energy  is,  as  it  were,  done  up  into  bundles, 
and  the  energy  in  any  particular  bundle  does  not  change  as 
the  bundle  travels  along  the  line  of  force.  Thus,  if  we 
consider  light  falling  upon  a  metal  plate,  if  we  increase  the 
distance  of  the  source  of  light,  we  shall  diminish  the  number 
of  these  different  bundles  or  units  falling  on  a  given  area  of 
the  metal,  but  we  shall  not  diminish  the  energy  in  the  in- 
dividual units ;  thus  any  effect  which  can  be  produced  by  a 
unit  by  itself,  will,  when  the  source  of  light  is  removed  to  a 
greater  distance,  take  place  less  frequently  it  is  true,  but 

1  Thomson  assumes  that  the  ether  has  disseminated  through  it  discrete 
lines  of  electric  force,  these  being  in  a  state  of  tension.  The  light  consists 
of  transverse  vibrations  travelling  along  these  lines. 


THE  PHOTO-ELECTRIC  EFFECT  495 

when  it  does  take  place  it  will  be  of  the  same  character  as 
when  the  intensity  of  the  light  was  stronger.  This  is,  I  think, 
the  explanation  of  the  remarkable  result  discovered  by  Lenard, 
that  though  the  number  of  corpuscles  emitted  by  a  piece  of 
metal  exposed  to  ultra-violet  light  increases  [as  intensity 
increases],  the  velocity  with  which  individual  corpuscles  come 
from  the  metal  does  not  depend  upon  the  intensity  of  the 
light.  If  this  result  stood  alone,  we  might  suppose  that  it 
indicated  that  the  forces  which  expel  the  corpuscles  from  the 
metal  are  not  the  electron  forces  in  the  light  wave  incident  on 
the  metal,  but  that  the  corpuscles  are  ejected  by  the  explosion 
of  some  of  the  molecules  of  the  metal  which  have  been  put 
into  an  unstable  state  by  the  incidence  of  the  light  •  if 
this  were  the  case  the  velocity  of  the  corpuscle  would  be 
determined  by  the  properties  of  the  atom  of  the  metal,  and 
not  by  the  intensity  of  the  light,  which  merely  acts  as  a 
trigger  to  start  the  explosion.  Some  experiments  made  quite 
recently  by  Dr.  E.  Ladenburg  make,  however,  this  last  ex- 
planation exceedingly  improbable.  Ladenburg  has  investi- 
gated the  velocities  of  corpuscles  emitted  under  the  action  of 
ultra-violet  light  of  different  wave  lengths,  and  finds  that  the 
velocity  varies  continuously  with  the  frequency ;  according  to 
his  interpretations  of  his  experiments,  the  velocity  is  directly 
proportional  to  the  frequency.  Thus,  though  the  velocity  of 
the  corpuscles  is  independent  of  the  intensity  of  the  light,  it 
varies  in  apparently  quite  a  continuous  way  with  the  quality 
of  the  light ;  this  would  be  very  improbable  if  the  corpuscles 
were  expelled  by  an  explosion  of  the  molecule.  It  seems 
more  reasonable  to  suppose  that  the  velocity  is  imparted  by 
the  light,  and  yet,  as  we  have  seen,  the  velocity  is  indepen- 
dent of  the  intensity  of  the  light.  These  results  can,  however, 
be  reconciled  by  the  view  stated  above,  that  a  wave  of  light 
is  not  a  continuous  structure,  but  that  its  energy  is  concen- 
trated in  units  (the  places  where  the  lines  of  force  are  dis- 
turbed), and  that  the  energy  in  each  of  these  units  does  not 
diminish  as  it  travels  along  its  line  of  force.  Thus  if  a  unit 
by  impinging  on  a  molecule  can  at  any  place  make  it  liberate 
a  corpuscle,  it  will  do  so  and  start  the  corpuscle  with  the 


496       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 

same  velocity,  whatever  may  be  the  distance  from  the  source 
when  it  strikes  the  "molecule;  thus  the  velocity  of  the  cor- 
puscles would  be  independent  of  the  intensity  of  the  light. 
Ladenburg  found  that  the  velocity  of  the  corpuscle  increases 
with  the  frequency  of  the  light  ;  this  shows  that,  if  the  view 
we  are  discussing  is  correct,  the  energy  in  the  units  will  increase 
with  the  frequency."  This  final  statement  is  simply  Planck's 
expression,  e  =  hv.  Sir  J.  J.  Thomson  has  likewise  made  an 
attempt  to  calculate  the  amount  of  energy  in  each  unit  of 
light  of  given  frequency  from  some  of  Lenard's  data.  With 
the  ultra-violet  light  used  by  Lenard  the  maximum  velocity 
of  the  corpuscles  was  about  io8  cms.  per  second;  the  kinetic 
energy  (&uvz)  of  a  corpuscle  moving  with  this  speed  is 
about  3  X  io-1'2  ergs.  If  we  suppose  that  the  energy  of  the 
corpuscle  liberated  is  of  the  same  magnitude  as  that  of  the 
light  unit  €  which  liberated  the  corpuscle,  we  would  expect  e 
(for  the  given  light)  to  be  about  io-1'2  ergs.  It  will  be  seen 
that  this  agrees  very  well  with  the  values  for  €  already  given 
as  a  result  of  calculation,  with  the  help  of  Planck's  constant  h. 
A  similar  calculation,  with  equally  satisfactory  results,  has 
been  recently  carried  out  by  J.  Franck  and  G.  Hertz  (Verh.  d. 
d.  physlk.  Gescll.)  13,  967,  1911;  ibid.,  14,  167,  1012).  The 
problem  of  the  probable  structure,  constitution,  and  origin 
of  these  light  units,  as  well  as  a  criticism  of  Planck's  views, 
has  been  considered  by  Sir  J.  J.  Thomson  (Phil.  Mag.>  1908- 
1910),  but  it  would  be  quite  out  of  the  scope  of  this  book  to 
attempt  anything  further  than  this  reference. 

To  show  by  an  extreme  case  how  few  light  units  it  is 
necessary  to  postulate,  say,  in  the  visible  region  of  the  spectrum 
under  ordinary  conditions,  Sir  J.  J.  Thomson  has  carried  out 
the  following  calculation.  Light  of  such  intensity  that  io~4  ergs 
fall  on  unit  area  per  second  would  be  very  faint,  but  would 
still  be  visible.  If  we  think  of  a  cylinder  of  ether,  base  i 
square  centimetre  and  3  X  io10  cms.  in  length,  then  the  above- 
mentioned  intensity  is  such  that  io~4  ergs  are  distributed 
throughout  this  cylinder  (at  any  moment),  and  the  density 


io 


-4 


of  the  light  is  therefore  -— ^  ergs  per  c.c.     Taking  as  a 


SPECIFIC  HEAT  OF  SOLIDS  497 

mean  value  for  one  quantum  the  number  io~12  ergs,  we  see 
that  there  will  only  be  one  such  unit  in  every  1000  c.c.  of  space. 
The  structure  of  light  is  therefore  of  a  coarse-grained  character. 
A  further  point  still  remains  to  be  mentioned.  The  idea 
of  a  point  source  not  radiating  uniformly  (this  being  a 
result  of  the  heterogeneous  nature  of  emission)  seems  to  be 
in  disagreement  with  the  ordinary  laws  of  propagation  of  light 
equally  in  all  directions.  It  must  be  remembered,  however, 
that  any  physical  source  of  radiation  consists  of  a  very  large 
number  of  resonators,  so  that  the  total  radiating  effect  is  on 
the  average  symmetrical  about  the  source. 

The  most  serious  drawback  to  the  unitary  theory  of  radiation 
is  to  be  found  in  the  phenomenon  of  interference,  especially 
interference  where  the  difference  of  path  is  very  great,  such  as 
it  was  —  about  40  cms.  —  in  some  measurements  of  Michelson. 

THE  THEORY  OF  SPECIFIC  HEAT. 

Einstein's  Extension  of  Planck's  Unitary  Theory  of  Energy  to 
the  calculation  of  Specific  Heats  of  Solids  (Crystalline  Sub- 
stances) and  Supercooled  Liquids  ("  Amorphoiis  solids"). 
(Cf.  Einstein,  Annalen  der  Physik,  [4]  22,  180,  1907.) 

The  specific  heat  of  a  substance  at  constant  volume  is 
defined  as  the  increase  in  total  energy  when  the  substance 
rises  i°  in  temperature.  Let  us  take  as  our  unit  of  mass  the 
gram-mole  or  gram-atom  (in  the  case  of  monatomic  sub- 
stances), and  we  then  can  write  — 


NOTE.  —  U  here  stands  for  total  energy  possessed  by  the 
substance  at  a  given  temperature.  It  is  not  to  be  confused 
with  the  significance  attached  to  U  in  the  "  elementary 
thermodynamical  treatment"  (Chap.  I.  Part  II.),  in  which 
"  U  "  stood  for  decrease  in  total  energy  due  to  chemical 
reaction. 

Let  us  now  restrict  our  attention  to  solids  (or  supercooled 
liquids),  taking  as  particular  instances  the  metallic  elements. 
These,  as  we  have  seen,  are  regarded  as  monatomic,  so  that  the 
T.P.C.  —  ii.  2  K 


498       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

gram-atom  and  gram-molecule  are  identical  terms  in  these  cases. 
Now  we  want  to  find*out  to  what  the  total  or  internal  energy  of 
a  metal  is  due.  It  is  usual  to  regard  the  solid  state  as  character- 
ised by  vibrations  of  the  atoms  about  their  respective  centres  of 
gravity.  Such  vibrations  can,  of  course,  take  place  in  the  three 
dimensions  of  space,  i.e.  each  atom  possesses  three  degrees  of 
freedom.  As  we  have  seen,  each  vibration  represents  energy 
one-half  of  which  is  kinetic,  one-half  potential,  as  long  as  the 
amplitude  of  the  vibration  is  not  too  great.  This  vibrational 
energy  is  regarded  as  representing  all  the  internal  energy 
possessed  by  the  atom,  at  least  at  low  temperatures  (at  high 
temperatures  the  energy  of  vibration  of  the  electrons  inside 
each  atom  would  have  to  be  considered,  but  at  ordinary 
temperature  and  at  lower  temperatures  the  total  energy  of  the 
solid  may  be  ascribed  to  the  vibration  of  the  atoms).  We 
have  already  discussed  this,  and  we  have  seen  that  on  applying 
the  principle  of  equipartition  of  energy  the  atomic  heat  of 
metals  should  be  3R  =  5-955  cals.  per  degree,  and  that  this 
should  be  independent  of  temperature.  As  already  pointed 
out,  this  numerical  value  is  certainly  approximated  to  at 
ordinary  temperatures,  but  instead  of  being  independent  of 
temperature,  it  varies,  becoming  continuously  smaller  as  the 
temperature  is  lowered.  The  question  therefore  which  arises 
is  how  this  variation  with  temperature  is  to  be  accounted  for. 
Einstein  in  1907  made  the  first  successful  attempt  at  the  solu- 
tion of  this  problem  by  suggesting  that  Planck's  unitary  theory 
of  energy  —  which  Planck  himself  had  applied  with  so  much 
success  to  the  problem  of  the  emission  of  radiant  energy  — 
could  also  be  applied  to  the  vibrational  energy  of  the  atoms, 
i.e.  to  the  total  internal  energy  of  the  solid,  the  temperature 
coefficient  of  which  is  identical  with  the  specific  or  atomic 
heat  of  the  substance  in  question  :  Planck's  expression  for  the 
energy  of  vibration  of  a  linear  resonator  (i.e.  an  atom  vibrating 
along  one  of  the  dimensions  of  space)  is,  as  we  have  seen  — 

u- 
•* 


The  energy  of  vibration  of  an  atom  capable  of  vibrating  along 


SPECIFIC  HEAT  OF  SOLIDS  499 

the  three  dimensions  of  space  will  be  three  times  this  quantity, 
and  if  we  denote  this  total  vibrational  energy  per  gram-atom  by 
U,  we  get— 

U=      M 

The  significance  of  U  is  identical  with  that  which  has  been 
ascribed  to  it  in  the  chapter  on  more  advanced  thermo- 
dynamics, namely,  the  total  energy  per  mole  or  gram-atom. 
In  the  case  considered  one-half  of  U  is  kinetic,  one- 
half  potential  energy.  N  denotes  the  number  of  atoms  in 
one  gram-atom.  On  this  view  the  vibrational  energy 
possessed  by  each  atom  must  be  an  even  multiple  of  one 
quantum.  On  the  older  view  ("  structureless  energy,"  so  to 
speak)  we  should  say  that  all  possible  differences  in  energy 
content  would  manifest  themselves  in  a  system  made  up  of 
a  large  number  of  vibrating  particles.  On  applying  the 
unitary  theory  of  energy  we  must  recognise  that  a  number  of 
atoms  have  no  vibrational  energy  at  all,  t.e.  are  at  rest.  Of 
those  vibrating,  the  energy  content  cannot  fall  below  the 
quantum  c',  where  e'  is  three  times  Planck's  quantum  e.  We 
have,  therefore,  sets  of  atoms  containing  energy  of  the  follow- 
ing amounts : — 

o,  c',  2e',  36',  and  so  on. 

In  order  to  bring  the  expression  for  U  given  above  into 
the  form  used  by  Einstein,  Nernst,  and  others,  we  shall  make 
a  slight  change  in  the  symbols.  If  we  denote  the  ratio  of 
Planck's  two  fundamental  constants  k  and  h  by  /30,1  we  can 

h  R 

write  -  =  j30  —  4*87  X  lo"11  C.G.S.  units.     Also,  since  k  =  ^, 

K  J-N 

R 
where  R  =  1*985  calories,  we  can  write  /iv  =  ^p0v. 

The  expression  for  U  then  becomes — 


1  This  is  frequently  written  as  jS.  The  slight  change  is  here  introduced 
to  prevent  any  confusion  with  (3,  one  of  the  terms  in  Nernst's  "heat 
theorem  "  equations  of  A  and  U. 


5oo       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 


It  will  be  observed  that  this  equation  differs  from  the 
expression  3RT  (obtained  by  applying  the  equipartition  prin- 
ciple, energy  being  regarded  as  continuous),  by  the  substitution 

of  the  term  -^Jff__  -  in  place  of  T. 

Differentiating  U  with  respect  to  T  we  obtain  Einstein's 
equation  for  the  atomic  (or  molecular)  heat  of  a  solid  at 
constant  volume,  viz.  — 


On  the  older  view,  the  factor  multiplied  by  $R  would  have 
been  unity.     This  "  correction  "  term  will  be  seen  to  contain 


o-s 
FIG.  94. 


i-o 


T  and  is  therefore  a  function  of  temperature.  On  Einstein's 
theory  one  would  expect  the  atomic  heat  itself  to  be  a  function 
of  temperature,  as  is  experimentally  the  case.  Qualitatively, 
therefore,  this  marks  a  considerable  advantage  over  the  older 
theory.  It  still  remains  to  be  seen  whether  the  expression 
given  really  reproduces  the  values  of  Ct>  at  various  tempera- 
tures quantitatively.  The  shape  of  the  curve  for  — -  as  given  by 

«T 

Einstein's  formula  is  shown  in  the  accompanying  figure  (Fig.  94) 
in  which  the  ordinates  represent  atomic  heats  and  the  abscissae 

T 

the  temperature  expressed  in  terms  of  5—.     For  any  given 

substance  the  vibration  frequency  is  taken  to  be  a  constant 


SPECIFIC  HEAT  OF  SOLIDS  501 

independent  of  the  temperature.  In  some  cases — in  salts  for 
example — where  there  is  more  than  one  characteristic  vibration 
frequency,  it  is  necessary  to  sum  a  number  of  similar  expres- 
sions, the  most  general  form  of  the  expression  being  given 


For  the  present,  however,  we  shall  use  the  expression  in  the 
simpler  form  given  above.     It  will  be  seen  from  the  figure  that 


T 

when  -A— •  >  0*9  the  term  /  p  ,,/Jj) V  "  w  approximates  to  unity, 

so  that  the  atomic  heat  becomes  equal  to  R.  Dulong  and 
Petit's  Law  is,  therefore,  a  consequence  of  Einstein's  theory 
when  the  temperature  is  not  too  low.  This  region  of  tempera- 
ture is  evidently  reached  at  room  temperature  in  the  case  of 
the  majority  of  metallic  elements.  As  in  radiation  phenomena 
we  see  that  the  equipartition  principle  applied  in  the  ordinary 
way  yields  results  (in  connection  with  atomic  heat)  which  are 
in  agreement  with  experiment  when  the  temperature  reaches 
a  certain  magnitude.  Einstein  tested  his  equation  not  on  the 
data  available  in  the  case  of  a  metal  but  on  the  diamond. 
Some  of  the  results  are  given  in  the  subjoined  table.  They  are 
also  indicated  by  circles  in  the  figure.  Einstein  chose  the 
experimentally  determined  value  for  Cv  at  T==33i*3°  and 
hence  calculated  v,  using  the  value  so  obtained  to  calculate 
the  values  of  Cv  at  other  temperatures.  The  experimental 
data  (quoted)  refer,  as  a  matter  of  fact,  to  Cp,  i.e.  the  atomic 
heat  at  constant  pressure.  In  the  case  of  the  diamond — 
though  not  so  in  other  cases — the  difference  between  Cp  and 
Ct,  is  small.  In  the  following  paragraph  we  shall  consider 
the  question  of  the  independent  determination  of  v  and  Cv 
respectively. 


502       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 


ATOMIC  *!EAT  OF  CARBON  (DIAMOND). 


T  (absolute). 

™, 

CD  calculated  from 
Einstein's  equation. 

Cp  observed 
(Weber). 

222'4 

0-1679 

0-762 

0*76 

262-4 

0-1980 

I-I46 

I'M 

2837 

0*2141 

1*354 

i  -35 

306*4 

0-2312 

I-582 

1*58 

331*3 

0-2500 

1-838 

1*84 

358*5 

0-2705 

2-II8 

2'12 

4I3*0 

0*3117 

2-661 

2-66 

479*0 

0-3615 

3-280 

3-28 

520*0 

0-3924 

3-63I 

3*63 

879-7 

0-6638 

5-290 

5*29 

1079-7 

0-8147 

5-387 

5]39 

1258-0 

0-9493 

5-507 

Measurements  of  Cv  of  the  diamond  at  low  temperatures 
down  to  30°  abs.  have  been  carried  out  by  Nernst,  who  dis- 
covered the  remarkable  fact  that  the  thermal  capacity  of  this 
substance  tends  practically  to  zero,  even  at  the  temperature 
+  50°  abs.  Between  this  temperature  and  absolute  zero, 

Cv  or  ^~;  =  o.     We  have  already  had  occasion  to  point  this 

out  in  connection  with  Nernst's  heat  theorem.  From  the 
shape  of  Einstein's  curve  (Fig.  94),  it  will  be  seen  that  C«  is 
is  tending  towards  zero  at  a  temperature  higher  than  o°  abs. 
It  thus  appears  that,  as  far  as  the  diamond  is  concerned, 
Einstein's  theory  reproduces  experimental  values  with  very 
considerable  fidelity.  While  giving  full  weight  to  such  general 
agreement,  it  is  also  necessary  to  point  out  that  this  agree- 
ment is  far  from  being  complete  in  many  other  cases.  We 
shall  return  to  this  after  having  described  in  outline  the  experi- 
mental methods  of  determining  Cr,  and  the  characteristic 
vibration  frequency  v. 


SPECIFIC  HEAT  OF  SOLIDS  503 


Experimental  Measurements  of  the  Specific  heats  of  Solids •, 
especially  at  Low  Temperature. 

(Cf.  Nernst,  Journ.  de  Physique^  [4]  9,  1910;  Nernst,  Koref 
and  Lindemann,  Sitzungsber.  Berl.  Akad.^  1910,  vol.  i,  p.  247  ; 
ibid.  Nernst,  p.  262  ;  Nernst,  Annalen  der  Physik.^  [4]  36,  395, 
1911.) 

One  form  of  calorimeter  consisted  of  a  heavy  vessel  of 
copper  (about  400  grams  in  weight),  the  good  thermal  con- 
ductivity of  such  a  mass  of  copper  doing  away  with  the  necessity 
of  stirring  the  substance,  which  is,  of  course,  impossible  in  the 
case  of  solids.  This  was  enclosed  in  a  Dewar  vacuum  vessel, 
placed  in  a  bath,  the  temperature  of  the  calorimeter  being 
measured  by  means  of  thermo-couples.  The  substance  to  be 
investigated  was  heated  or  cooled  to  a  known  temperature, 
and  introduced  into  the  calorimeter,  the  change  in  tempera- 
ture of  which  was  observed.  Knowing  the  heat  capacity  of  the 
calorimeter,  the  specific  heat  of  the  substance  could  be  obtained. 
This  method  worked  admirably,  but  of  course  it  is  limited  to 
the  determination  of  mean  values  of  Cv,  holding  over  a  con- 
siderable temperature  range.  For  the  purposes  in  view,  it 
was  necessary,  however,  to  be  able  to  determine  C»  for  small 
temperature  ranges,  that  is  for  a  consecutive  series  of  "  points  " 
on  the  temperature  scale.  To  accomplish  this,  a  different 
procedure  had  to  be  adopted. 

The  principle  of  this  second  method  consists  briefly  in 
making  the  investigated  substance  itself  act  as  its  own  calori- 
meter. The  substance  could  be  heated  electrically  by  means 
of  a  platinum  spiral,  through  which  a  known  quantity  of 
electrical  energy  was  passed,  as  measured  by  an  ammeter  and 
voltmeter  in  the  circuit.  The  rise  in  temperature  of  the 
substance  was  obtained  by  using  the  heating  spiral  itself  as  a 
resistance  thermometer,  i.e..  its  resistance  was  observed  by 
means  of  a  bridge,  before  and  after  the  heating  of  the  substance, 
the  alteration  in  resistance  giving  the  temperature  change. 
Knowing  the  mass  of  substance  employed,  the  electrical  energy 
supplied,  and  the  rise  in  temperature,  it  is  easy  to  calculate 


504       A   SYSTEM  OF  PHYSICAL   CHEMISTRY 


the  specific  heat  of  the  substance.  This  extremely  ingenious 
method  was  worked  out  by  Eucken,  in  Nernst's  laboratory 
(Physik.  Zeitschr.^  10,  586,  1909),  and  has  been  employed  by 
Nernst  and  his  collaborators  for  the  accurate  determination  of 
specific  heats  of  substances  at  various  temperatures,  extending 
over  a  wide  range,  even  down  to  the  temperature  of  liquid 
hydrogen.  A  few  details  may  be  given  here.  A  pear-shaped 
vessel  of  glass  (shown  in  the  accompanying  figure,  Fig.  95) 
contains  the  calorimeter  K  suspended  by  the  wires  which  are 
connected  to  the  heating  spiral.  By  means  of  a  Gaede  pump, 


\     / 


K 


FIG.  95. 

together  with  a  vessel  of  charcoal  cooled  in  liquid  air,  a  very 
good  vacuum  is  maintained  in  the  glass  vessel,  so  as  to 
prevent  any  loss  of  heat  by  convection.  The  calorimeter  K 
(i.e.  the  substance  itself),  together  with  the  connecting  wires,  is 
brought  to  a  certain  temperature,  which  is  rapidly  brought 
about  by  substituting  hydrogen  gas  in  place  of  air  (since  the 
hydrogen  conducts  thermally  so  much  better).  The  hydrogen 
is  subsequently  removed,  the  vessel  being  evacuated.  The 
spiral  resistance  is  of  purest  platinum.  Its  change  in  resist- 
ance with  temperature  was  calibrated  by  comparison  with  an 


SPECIFIC  HEAT  OF  SOLIDS  505 

oxygen  gas  thermometer.  The  calorimeter  K  varied  in  con- 
struction, according  as  to  whether  the  substance  investigated 
conducted  heat  well  or  badly.  The  first  type  shown  in  the 
figure  is  suitable  when  the  substance  is  a  metal.  It  consists 
of  a  small  cylinder  of  the  metal,  having  a  cylindrical  hole 
drilled  almost  throughout  its  length.  Into  this  a  plug  of  the 
same  metal  fits  loosely,  the  heating  spiral  occupying  the  space 
between.  To  insulate  the  spiral  thin  paraffined  paper  was 
employed,  liquid  paraffin  itself  being  finally  poured  in.  The 
upper  part  of  the  plug  is  somewhat  thicker  than  the  remainder, 
good  thermal  contact  being  thus  obtained.  For  substances 
which  conduct  heat  badly  the  second  form  of  calorimeter 
was  employed.  A  vessel  of  silver  was  used,  the  platinum 
spiral  being  wound  round  a  cylindrical  sheet  of  silver,  which 
served  to  spread  the  heat  rapidly.  Electrical  insulation  of  the 
spiral  was  maintained  by  the  help  of  shellac  and  thin  silk. 
The  platinum  wire,  on  entering  the  calorimeter,  passes  through 
a  small  insulating  tube,  and  finally,  after  passing  through  the 
body  of  the  calorimeter,  ends  on  the  silver  vessel,  to  which  a 
lead  is  attached.  It  was  found  that  some  air  must  be  left  in 
the  calorimeter  itself  (not  in  the  pear-shaped  vessel),  so  as  to 
ensure  better  thermal  conduction,  and  attainment  of  equili- 
brium. Nernst  also  describes  a  third  form  of  calorimeter, 
suitable  for  liquids,  details  of  which  will  be  found  in  the  paper 
cited.  The  heat  capacity  of  the  fittings  of  the  calorimeter  K 
— other  than  the  actual  substance  examined — was  determined, 
and  found  to  agree  well  with  that  calculated  from  the  known 
masses  and  specific  heats  of  the  substances  employed.  When 
the  pear-shaped  vessel  was  well  evacuated,  the  temperature 
of  the  substance,  i.e.  the  calorimeter  K  (as  measured  by  the 
resistance  of  the  spiral)  was  found  to  remain  remarkably 
constant  for  a  considerable  time,  even  when  K  had  been 
previously  cooled  down  to  very  low  temperatures,  by  placing 
the  non-evacuated  pear-shaped  vessel  in  contact  with  liquid 
air.  Similarly,  after  cutting  off  the  current  the  new  (final)  tem- 
perature was  again  found  to  remain  constant,  and  be  accurately 
measurable.  Nernst  considers  that  the  probable  error  in  the 
results  does  not  exceed  i  per  cent.,  and  may  be  much  less. 


506       A    SYSTEM  OF  PHYSICAL    CHE.IflSTRY 

Some  of  -the  experimental  data,  obtained  by  the  above 
means,  have  already  Been  given  in  dealing  with  Xernst's  heat 
theorem,  and  other  examples  will  be  given  when  we  take  up 
in  greater  detail  the  question  of  the  applicability  of  Einstein's 
expression  to  the  atomic  heat  of  solids  in  general. 

METHODS  OF  DETERMINING  THE  CHARACTERISTIC  VIBRATION 
FREQUENCY  OF  A  SOLID. 

First  Method. 

This  depends  on  the  direct  measurement  of  the  4i  Rest- 
strahlen  (Residual  Rays)."  Many  substances  possess  the 
property  of  selective  reflection,  that  is  they  powerfully  reflect 
rays  of  certain  wave-length.  The  rays  which  are  most  strongly 
reflected  in  the  case  of  incident  light  are  those  which  are  most 
strongly  absorbed  when  the  light  is  transmitted.  The  wave- 
lengths strongly  absorbed  are  identical  with  the  characteristic 
wave-lengths  corresponding  to  the  natural  vibration  frequencies 
of  the  substance,  for  it  is  when  the  light  has  the  same  frequency 
as  that  of  the  vibrating  atom  that  it  is  most  strongly  absorbed. 
If  a  beam  of  incident  light  of  all  wave-lengths,  that  is  to  say 
a  beam  emitted  by  a  heated  black  body,  is  reflected  succes- 
sively from  a  number  of  pieces  of  the  solid  in  question,  at 
each  reflection  the  beam  will  become  purer,  that  is  the  re- 
flected light  tends  more  and  more  to  consist  of  the  vibra- 
tion frequency  characteristic  of  the  solid.  The  rays  which 
survive  after  having  suffered  a  number  of  such  reflections 
are  called  the  "  Residual  Rays."  The  measurement  of  the 
wave-length  and  the  energy  of  such  Reststrahlen  has  been 
principally  carried  out  by  Prof.  Rubens  of  Berlin  and  his  col- 
laborators, although  the  idea  was  that  of  Beckmann  in  the 
first  instance  (cf.  Rubens  and  Nichols  Annalen  der  Physik., 
60,  418,  1897  ;  Rubens  and  Aschkinass,  ibid..  65,  241,  1898  : 
Rubens,  ibid.,  69,  576,  1899;  Rubens  and  Kurlbaum,  ibid., 
[4]  4,  649,  1901  ;  Rubens  and  Hollnagel,  Phil.  Mag.,  [6]  19, 
761,  1910.)  These  rays  corresponding  to  the  vibration 
frequency  of  atoms  are  naturally  far  in  the  infra-red  region. 
The  apparatus  employed  is  shown  diagrammatically  in  the 


V1BRATIOX  FREQUENCY  OF  SOLIDS         507 


figure  (Fig.  96).  L  is  the  electrically  heated  black  body 
emitting  waves  of  all  frequencies.  D!  and  D2  are  diaphragms 
of  small  aperture.  S  is  a  screen.  The  diaphragms  and  screen 
are  kept  cool  with  water  at  room  temperature.  The  substance 
under  examination  is  represented  by  Pj,  P2,  PS,  ¥4.  The 
surfaces  of  these  pieces  are  polished  as  highly  as  possible.  M 
is  an  adjustable  mirror  and  T  the  thermopile.  A  diffraction 
grating  of  special  make  could  be  inserted  in  the  path  of  the 
beam,  thereby  analysing  it  into  a  spectrum,  the  position  of  the 
energy  maxima  corresponding  to  the  characteristic  vibration 
frequencies  of  the  substance  P  being  determined  by  means  of 
the  thermopile.  In  later  experiments  the  grating  had  to  be 

D2 


FIG.  96. 

abandoned  owing  to  the  large  amount  of  absorption  suffered 
by  the  rays  in  passing  through  it.  Instead,  an  interference 
method  has  been  devised — see  the  paper  by  Rubens  and 
Hollnagel,  Phil.  Mag.,  I.e.  The  following  table  contains  some 
of  the  results  summarised  by  Rubens  and  von  Wartenberg 
(Sitzungsber.  konigl.  preuss.  Akad.y  1914,  p.  169).  In  actual 
experiment  two  bands  are  usually  observed  fairly  close  together. 
One  of  these,  however,  is  due  to  the  presence  of  water  vapour 
(Rubens,  ibid.,  1913,  p.  513)- 

RESIDUAL  RAYS  CHARACTERISTIC  OF  CERTAIN  SOLID  SALTS. 


Substance. 

Wave-length  in  /*. 

Substance. 

Wave-length  in  H. 

H4C1  . 
aCl      . 

SI'S 
C2'0 

KI        .      . 
CaCOs      . 

94- 

Cl.      .         i          63-4 
gCl     .                   81-5 

AgBr    .      .            112-7 
TlBr     .      .             117-0 

Br       .         j           82-6 

Til       .      .            I5i'8 

bCL    .         |          91-0 

5o8       A   SYSTEM  OF  PHYSICAL    CHEMISTRY 

•    Second  Method, 

This  is  an  indirect  method  due  to  Einstein  (Annalen  d. 
Physik.)  [4]  34,  170,  1911).  From  consideration  of  the  dis- 
tribution of  atoms  in  metals  and  the  forces  acting  upon  them 
when  they  are  displaced  from  their  centres  of  gravity  during 
vibration,  Einstein  obtained  an  approximate  expression  for 
the  characteristic  vibration  frequency  v  in  terms  of  the  com- 
pressibility K,  the  atomic  (or  molecular)  volume  V,  and  the 
atomic  (or  molecular)  weight  M,  which  may  be  written  as 
follows : — 

V« 


Using  Griineisen's  recent  data  for  the  compressibility  (Ann. 
der  Physik.,  25,  848,  1908),  Einstein  calculated  the  following 
characteristic  vibration  frequencies  : — 

SOME  OF  EINSTEIN'S  VALUES  FOR  v. 


Substance. 

v. 

Substance. 

, 

Al 
Cu 

6-6  x  io12 

57 

Fe     . 
Ft      . 

. 

6-5 

4-6 

Ag 

4*1 

Pb     . 

2-2 

Au 

3'8 

Cd     . 

2-6 

Ni 

6-6 

Bi      . 

1-8 

Einstein  himself  does  not  claim  that  this  method  is  more 
than  a  rough  approximation.  Nevertheless  the  values  agree 
moderately  well  with  those  obtained  by  other  methods.  At 
most  the  method  only  gives  a  mean  value  of  the  frequency 
even  in  cases  in  which  there  may  be  more  than  one  charac- 
teristic frequency. 


Third  Method. 

This  method,  also  an  indirect  one,  is  due  to  F.  A.  Linde- 
mann  (Physikal.  Zeitsch.^  11,  609,  1910).  It  is  based  upon  the 
assumption  that  at  the  melting  point  Ts  of  a  solid  the  amplitude 


VIBRATION  FREQUENCY  OF  SOLIDS          509 

of  vibration  of  the  atoms  is  approximately  equal  to  the  mean 
distance  of  the  atoms  apart.  Let  us  denote  by  r8  the  radius  of 
the  circular  vibration  of  an  atom  (in  a  metal,  say),  then  the 
mean  velocity  u  with  which  it  vibrates  is  given  by  u  =  27rrsv, 
where  v  is,  of  course,  the  number  of  such  vibrations  per  second. 
The  mean  kinetic  energy  of  vibration  is  \mu*,  where  m  is 
the  mass  of  the  atom,  and  since  there  is  likewise  present  an 
equal  quantity  of  potential  energy,  the  total  mean  energy  of 
a  vibrating  atom  is  mu2-  or  4m7T2r82v2.  Now  at  the  melting 
point  the  metals  may  be  considered  as  very  closely  obeying 
Dulong  and  Petit's  Law,  i.e.  their  atomic  heat  is  practically  6. 
That  is,  the  expression  for  the  vibrating  energy  which  is  given 


N 
for  large  values  of  T  as  already  pointed  out. 

Hence 


If  we  now  regard  the  atomic  volume  V  as  proportional  to 
the  cube  of  the  mean  distance  apart  of  the  atoms  (which 
distance  is  identical  with  rs  on  Lindemann's  assumption),  we 
can  write — 

v  oc 


The  proportionality  factor  obtained  by  comparison  with 
experiment  is  2*80  X  io12,  and  hence  Lindemann's  formula 
is — 

v=  2-80  X 


It  is  thus  possible  to  calculate  v  from  measurements  of 
melting  temperature,  molecular  weight  of  the  atoms,  and  the 
atomic  volume.  This  is  an  extremely  convenient  method, 
but  has  the  disadvantage  of  not  giving  v  very  accurately.  It 


510       A   SYSTEM  OF  PHYSICAL    CHEMISTRY 

appears,  however,  to  be  somewhat  more  accurate  than  Ein- 
stein's method  (Method  2).  The  following  table  is  given  by 
Nernst  and  Lindemann  (Z.  Elektrochem.^  17,  822,  1911).  The 
observed  values  of  v  are  those  given  by  a  formula  of  Nernst 
and  Lindemann  which  is  a  modification  of  Einstein's  expres- 
sion for  atomic  heat,  which  at  the  same  time  represents  experi- 
mental values  more  closely.  We  shall  discuss  this  expression 
later,  but  in  the  meantime  it  may  be  taken  as  giving  the  values 
of  v  from  atomic  heat  measurements. 

LINDEMANN'S  FORMULA  FOR  "v." 


Substance. 

Molecular 
weight. 

T« 
melting- 
point. 

V 

molecular 
volume. 

v 
calculated  by  Lin- 
demann's  formula. 

v 

"observed." 

Al 

23-I 

930 

I0'0 

7'6  x  io12 

8-3  x  io12 

Cu 

63-6 

1357 

7'I 

6-8 

6-6 

Zn 

65-4 

692 

9-2 

4'4 

4-8 

Ag 

107-9 

1234 

10-3 

4'4 

4*5 

Pb 

206  -9 

600 

18-3 

1-8 

1-9 

Diamond 

I2'0 

3600? 

3'4 

32-5 

40-0 

Iodine 

I27'O 

386 

257 

17 

2'0 

NaCl 

29-2 

1083 

I3-5 

7-2 

5'9 

KC1  . 

37-2 

1051 

18-9 

5-6 

4'5 

Lindemann's  formula,  like  Einstein's,  gives  only  a  single 
mean  characteristic  vibration  frequency.  For  this  purpose, 
therefore,  the  molecule  of  NaCl,  KC1,  etc.,  is  treated  as  a 
single  atom  having  an  atomic  weight  one-half  of  that  usually 
assigned  to  the  molecule  of  these  substances. 

Of  the  three  methods  the  Residual  Ray  method  is  the 
most  accurate  and  reliable. 


The  System  of  Elements  and  the  Periodicity  of  the  Vibration 
Frequency  with  the  Atomic  Weights. 

(Cf.  W.  Biltz,  Zeitsch.  Elektrochem.,  17,  676,  1911.) 

Employing  Lindemann's  formula,  Biltz  has  calculated  the 
characteristic  vibration  frequency  v  for  the  elements  (in  the 
solid  state)  as  far  as  the  existing  data  upon  melting  point 


VIBRATION  FREQUENCY  OF  SOLIDS          511 

and  density  permitted,  and  has  traced  out  a  relation  between 
the  frequency  and  the  atomic  weight.  In  the  following  table 
are  given  the  values  of  v  for  each  element  as  calculated  by  the 
help  of  Lindemann's  formula  l : — 


Element. 

Density. 

Melting-point 
temperature 
T  absolute. 

Atomic  weight. 

v  X  io-12. 

H.     .      . 

0-0763 

14 

I  '01 

4-36 

He  (liquid) 

0-154 

Ca2 

3  "99 

0-66 

Li       .      . 

°*59 

459 

6-94 

I0'0 

Be       .      . 

173 

Ca  1200 

9-1 

18*5 

C  (graphite) 

2-3 

„  3600 

I2'p 

277 

C  (diamond) 

3*52 

„  3600 

12-0 

317 

N  .      ,      . 

1-03 

62-5 

14*0 

2'5 

0  .     ,     . 

28- 

16-0 

1-7 

F  (liquid) 

I'M 

50 

19-0 

1-8 

[Ne  (liquid) 

1-24 

Caz 

20'2 

0'34 

Na      .      . 

0-98 

371 

23-0 

3*9" 

Mg      .      . 

1-74 

924 

24-3 

7-2 

Al       .      . 

2-66 

930 

27-I 

7*5 

Si.     .     . 

2-49 

1700 

28-3 

9-6 

P(red)     . 

903 

31-0 

6-3 

S  (rhombic) 

2-07 

388 

32-1 

3-96 

Cl  (liquid) 

1-66 

171 

35*5 

2-24 

Ar  (liquid) 

1-42 

85 

39'9 

1-32 

K  . 

0-87 

335 

39'1 

2-3 

Ca 

i*59 

1050 

40'  i 

4'9 

Ti 

4'5o 

2120 

48-1 

8*4 

Vd 

5'8 

1950 

8'3 

Cr 

6-74 

1790 

52-0 

8*3 

Mn 

7'39 

1520 

54'9 

7'5 

Fe 

7-85 

1820 

ss;9 

8'3 

Co 

872 

1780 

8-2 

Ni 

8-90 

1700 

587 

8'2 

Cu 

8'93 

1357 

63-6 

6'  7 

Zn 

7-12 

692 

65-4 

4-36 

Ga 

5'95 

303 

69-9 

2'5 

Ge 

5*47 

Ca  i  loo 

72-5 

4'6 

As 

573 

„    1000 

75'° 

4'3" 

Se  (grey)  . 

4-8 

490 

79'2 

2-7 

Br  (liquid) 

266 

79-9 

17 

Kr  (liquid) 

2-16 

104 

82-9 

0-9 

Rb      .      . 

1-52 

311 

85'5 

i'45 

Sr        .      . 

2-63 

[1080] 

87-6 

3-0 

1  The  values  of  v  are  slightly  different  from  those  given  by  Biltz,  as 
Biltz  employed  the  older  and  less  correct  proportionality  factor  2' 12  x  io12 
in  Lindemann's  formula  instead  of  the  later  one,  2'8o  X  io12. 


512       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 


Element. 

Density. 

Melting-point 
temperature 
T  absolute. 

Atomic  weight. 

V  X  10    12. 

[Zr 

[Nb 

6-4 

(702700 
2220 

90-6 

93'5 

— 

Mo 

9'01 

2400 

96-0 

6*3 

[Ru 

I2'3 

Ca  2250 

I02-O 

— 

Pd 

II'9 

1860 

107-0 

574 

Ag 

10-5 

1234 

108-0 

4-36 

Cd 

8*65 

112*0 

3*0 

In             .    • 

7*12 

428 

115-0 

Sn 

7-30 

505 

119*0 

2-24 

Sb 

6*62 

904 

120-0 

3*o 

Te 

6-23 

723 

I27-5 

2-45 

I    . 

4-66 

386 

I27-0 

r6 

X  (liquid) 

133 

I30-0 

0-85 

Cs 

1*89 

299 

133-0 

0-95 

Ba 

378 

1  120 

I37-0 

2-4 

La 

6-16 

I085 

I39-0 

2-8 

Ce 

7*04 

896 

I40-2 

2-64 

Ta 

16-6 

2575 

ill 

475 

Os 

22-5 

2700 

191 

Ir  . 

22-4 

2500 

193 

4'9 

Pt 

2018 

195 

4-36 

Au 

I9-3 

1337 

197 

3'4 

Hg 

14-2 

234 

200 

1-25 

Tl 

n-8 

575 

204 

1-84 

Pb 

11*4 

600 

207 

1-84 

Bi 

978 

54i 

208 

r6 

fTh 

I2'2 

232 

3-2] 

Biltz  points-  out  that  in  the  characteristic  frequency  we 
have  a  very  fundamental  atomic  constant.  Further,  since  its 
measurement  involves  two  properties,  namely,  specific  volume 
and  melting  point,  it  is  a  step  further  than  the  stage  reached, 
for  example,  in  Lothar  Meyer's  curve,  in  which  one  property 
only  is  brought  into  relation  to  the  atomic  weight.  Some 
doubt,  of  course,  exists  regarding  the  values  in  the  case  of  a 
few  elements,  especially  those  possessing  allotropic  forms. 
Biltz  has  plotted  the  frequencies  against  the  atomic  weights, 
and  has  found  the  periodic  variation  shown  in  the  figure 
(Fig.  97).  It  will  be  seen  to  be  quite  analogous  to  Mende- 
leef's  original  scheme.  The  majority  of  the  elements  occupy 
places  on  the  curve,  exceptions  being  fluorine,  manganese, 
tin  and  tellurium.  Argon  and  potassium  occupy  as  usual  their 


VIBRATION  FREQUENCY  OF  SOLIDS          513 

anomalous  position.  The  position  of  hydrogen,  which  was  for 
a  long  time  in  doubt,  is  clearly  that  of  the  first  member  of  the 
halogen  series,  as  Ramsay  had  insisted  on  many  years  ago. 
The  unique  position  of  carbon,  with  extremely  high  value  of  V, 


w   Os 


V*ft 

,'       T  i 

\Au 

\TI  Pb 


Atomic     Weight, 
FIG.  97. 

brings  out  very  clearly  its  unique  chemical  character  as  regards 
its  immense  capability  of  forming  compounds.  An  atom 
possessing  high  frequency  no  doubt  is  the  most  likely  type  of 
atom  to  allow  of  the  transfer  of  electrons  to  and  from  itself,  i.e. 
valency  electrons. 


T.P.C.— II. 


2  L 


514       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 


Further  Consideration  of  Einstein's  Expression  for  Atomic  If  eat. 
Einstein's  expression,  viz.  — 


has  been  applied  to  several  metals  and  solid  salts.  The 
following  two  examples,  at  low  temperatures,  given  by  Nernst 
and  Lindemann  (Zeitsch.  Elektrochem.,  17,  818,  1911),  are 
instructive. 

COPPER.    /?0v  =  240. 


Temperature, 
absolute. 

Atomic  heat 
(observed). 

Atomic  heat 
(calculated}. 

86-0 

33'4 
22-5 

3-38 
0-538 
0-223 

3'3J 
0-234 
0'023 

POTASSIUM  CHLORIDE.    /30v  =  218. 


Temperature, 
absolute. 

Atomic  heat 
(observed). 

Atomic  heat 
(calculated). 

86-0 

4'36 

3'54 

52-8 

2-80 

1-70 

30-1 

0-98 

0-235 

22-8 

0-58 

0-039 

It  is  clear  that  while  Einstein's  formula  gives  the  qualita- 
tive course  of  the  variation  of  Cv  with  temperature,  it  is  far 
from  reproducing  the  variation  quantitatively.  It  might  be 
thought  that  this  is  due  to  an  error  in  using  a  single  frequency 
value  where  in  the  actual  case  there  might  be  a  broad  band. 
From  Rubens'  results  already  quoted  it  will  be  seen,  however, 
that  the  characteristic  frequency  of  each  salt  is  fairly  isolated, 
and  at  lower  temperatures  this  isolation  would  become  still 


SPECIFIC  (ATOMIC]    HEAT  OF  SOLIDS        515 

more  perfect.  In  metallic  elements  themselves,  where  it  is 
usually  regarded  that  we  are  dealing  with  uncombined  (un- 
polymerised)  atoms  themselves,  the  characteristic  frequency 
should  be  quite  sharp  and  well  defined. 

As  a  way  out  of  the  difficulty — prior  to  the  publication  of 
Debye's  formula  of  1912,  to  which  reference  will  be  made 
later — Nernst  and  Lindemann  proposed  an  empirical  expres- 
sion for  Cr>  which  differs  somewhat  from  that  of  Einstein. 


The  Nernst- Lindemann  Formula  for  the  Atomic  Heat  of  Solids. 
•  This  expression  takes  the  following  form  : — 


C«  = 


(i) 


that  is,  the  atomic  heat  expression  of  Einstein  has  been  split 
up  into  two  portions.  One  portion  represents  the  energy 
corresponding  to  the  characteristic  frequency  v;  the  second 
portion  represents  the  energy  due  to  the  harmonic  vibration, 
which  is  one  "  octave  "  lower  than  the  characteristic,  namely, 

- .  Why  this  "  octave  "  should  be  introduced  is  as  yet  practi- 
cally unexplained.  The  expression,  however,  reproduces  the 
experimental  value  of  CK  with  extraordinary  accuracy.  Its 
importance  is  therefore  not  to  be  questioned.  Some  illus- 
tration of  its  applicability  may  be  given  here. 

First  of  all,  we  have  to  notice  that  the  direct  experimentally 
determined  values  of  the  atomic  heat,  such  as  those  of  Nernst 
and  his  collaborators,  hold  for  constant  pressure,  i.e.  they  are 
(Cp)  values.  In  the  formula,  however,  it  is  the  atomic  heat  at 
constant  volume,  viz.  C«  which  occurs.  They  are  nearly 
identical,  but  not  quite.  We  have,  therefore,  to  convert  the 
one  into  the  other.  The  procedure  is  as  follows.  On  the 
basis  of  thermodynamics  we  have  the  relation — 


516       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 


where  a  is  the  linear  coefficient  of  expansion,  V  the  molecular 
or  atomic  volume,  and.  K  the  compressibility.  Now  experi- 

V 

ment  has  shown  that  it  is  allowable  to  consider  „  as  a  constant, 

s\. 

independent  of  T  (especially  as  it  occurs  in  a  small  correc- 
tion term),  and  further,  making  use  of  Gruneisen's  relation 
(Verhand.  d.  d.  physik.  Gesell.,  1911,  491)  regarding  the 
proportionality  between  Q>  and  the  expansion  coefficient  a, 
we  can  rewrite  the  above  expresssion  in  the  form — 

Cp  =  C,  +  C|TA  i 

where  A  is  a  constant  characteristic  of  each  substance.  The 
values  of  A  are  given  in  the  following  table  for  some  metals 
and  salts : — 

VALUES  OF  THE  CONSTANT  "A." 


.Substance. 

Atomic  volume 
V. 

301  X  IOB. 

K  X  10'-. 

C,/<V 

A  X  10  . 

Al 

I0'0 

72 

1-48 

1-042 

2'2 

Cu 

7-1 

48 

0785 

1-025 

i'3 

5f 

10-3 

55 

0775 

1-047 

2-5 

Pb 

18-3 

82 

2'4 

1-055 

3-0 

Ft 

9-1 

27 

0-40 

1-019 

I'O 

NaCl 

13'S 

121 

4-28 

1-051 

2-7 

KC1 

18-9 

II4 

7-6 

1-038 

2-0 

In  the  case  of  substances  for  which  the  compressibility  K 
and  the  coefficient  of  expansion  a  are  known,  it  is  easy  to 
convert  Cr  values  into  Q>,  or  vice  versa.  For  other  substances, 
however,  the  procedure  is  somewhat  different.  The  individual 
constant  A  can  be  shown  to  be  inversely  proportional  to  the 
melting-point  Tfi  (cf.  Nernst  and  Lindemann,  I.e.).  We  can 

1  This  expression  has  been  recently  transformed  into  a  still  simpler 
one,  viz.  Cp  =  Cv  +  0T*,  where  a  is  an  empiric  constant  characteristic  of 
the  substance  in  question  (Lindemann  and  Magnus,  Z.  Elektrochein.,  16, 
269,  1910;  cf.  also  Nernst,  Ann.  Physik.^  36,  395,  1911).  Since  direct 
measurement  always  gives  us  C^,,  this  is  a  very  convenient  way  of 
getting  C,,. 


SPECIFIC   (ATOMICS    HEAT  OF  SOLIDS        517 

thus  write  A  =  TJ?>  where  A0  is  a  universal  constant  having  the 
value  0-0214  (C.G.S.  units).     Hence — 


p  =    _,„  --      <jp  Q 

-I  ,s' 

The  general  method  adopted  by  Nernst  and  Lindemann  to 
test  their  formula,  equation  i,  was,  first  of  all,  to  calculate  Cu 
by  its  means  for  substances  for  which  v  is  known  (either  by 
means  of  Reststrahlen  measurements,  or  by  means  of  Linde- 
mann's  melting-point  formula).  The  "calculated"  value  of 
C«  was  then  converted  into  terms  of  Q>  by  means  of  the 
expression — 

Q,  =  CV  +  C|TA 

and   the   result   finally   compared  with   the   observed   values 
of  Q,. 

The  following  tables  give  the  results  obtained  in  the  case 
of  aluminium,  copper,  silver,  lead,  and  the  salts  KG,  NaCl, 
KBr,  and  the  diamond.  These  are  all  taken  from  the  paper 
of  Nernst  and  Lindemann  (I.e.) : — 


ALUMINIUM. 


=  405. 


T  absolute. 

Qp  calculated. 

C    calculated. 

C    observed. 

Observer. 

32H 

0-23 

0-23 

0-25 

Nernst. 

35"1 

0-31 

0-3I 

0'33 

j» 

83-0 

2-42 

2'43 

2-41 

» 

86-0 
88-3 

2-52 
2'6l 

2-53 
2-62 

2-52 
2'62 

»  » 
ii 

137-0 

3-99 

4-05               3'97 

Koref. 

235-0 

33i'o 
433*o 

5-15 
5-52 
570 

5-30 
5-76 
6-06 

5^2 
I'82 

6'io 

Koref,  Schimpff. 
Magnus,  Schinipft". 
Magnus. 

555*0 

5-80 

6-30 

6-48 

it 

5i8        A    SYSTEM   OF  PHYSICAL    CHEMISTRY 


COPPER.    @0v  =  321. 


T  absolute. 

Cv  calculated. 

C    calculated. 

CL  observed. 

Observer. 

23-5 

0-15 

0-15 

0-22 

Nernst. 

27-7 

0-31 

0-31 

0-32 

ii 

33'4 

°'59 

0'59 

0-54 

ii 

87-0 

3'35 

3'37 

3'33 

u 

88-0 

3'37 

3-39 

3'38 

tt 

137-0 
234-0 

4-60 

5-52 

4'57 
5'59 

Koref. 
Koref,  Schimpff. 

290-0 
323-0 

5-60 
5-66 

575 
5-81 

579 
5-90      { 

Gaede. 

Bartoli,  Stracciati, 
Schimpff. 

450-0 

5-81 

6-03 

6-09 

Magnus. 

SILVER, 


=  221. 


T  absolute. 

C^  calculated. 

C    calculated. 

C    observed. 

Observer. 

35*0 

I'59 

i  '59 

I-58 

Nernst. 

39'  I 

I-92 

1-92 

I'90 

42-9 

2'22 

2'22 

2-26 

45'5 

2'43 

2-44 

2-47 

5*'4 

2-81 

2-82 

2-81 

53'8 

2'97 

2-98 

2-90 

77-0 

4-07 

4'II 

4-07 

lOO'O 

4-72 

477 

4-86 

Koref. 

200-0 
273-0 

33  1  'o 

5-60 

577 
5-82 

577 

6'02 

6-12 

578 
6-oo 

6-01       | 

Koref,  Schimpff. 
Bartoli,  Stracciati, 
Schimpff. 

535-o 
589-0 

5*90 
5-92 

6-45 

6'57 

6-46 
6-64 

Magnus. 
» 

SPECIFIC   (ATOMIC]    HEAT  OF  SOLIDS         519 


LEAD. 


=  95. 


T  absolute. 

Cv  calculated. 

C    calculated. 

C    observed. 

Observer. 

23-0 

2-95 

2-96 

2'96 

Nernst. 

28-3 

3-63 

3-64 

ii 

36-8 

4-35 

4'37 

4-40 

38-1          |          4'43 

4-45 

4*45 

M 

5-60 

5-68 

5-65 

,, 

90-2 

5-62 

5'70 

J} 

200*0 

5-9o 

6*12 

6-13 

Koref. 

273-0           1            5*92 

6-24 

6-31 

Koref,  Gaede. 

5-92               6-26 

6-33 

Gaede. 

332-0 
409-0 

5  '93 
5'94 

6-31 
6-40 

6-41 
6'6i 

Magnus,  Schimpff. 
Magnus. 

KC1. 


is  the  mean  of  the  two  values  232*4  and  203*2.' 


T  absolute. 

C,tf  calculated. 

Cp  calculated. 

CL  observed. 

Observer. 

22-8 

0-61 

0*6l 

0-58 

Nernst. 

26-9 

0-70 

0*70 

0-76 

„ 

30-1 

1*23                1*23                0*98? 

ii 

337 

i'53                ''53                1-25? 

ii 

39-0 

48-3 
52*8 

1*98 
2-66 
2*96 

1-98 

2*66 
2-97 

I|3 
2-80 

ii 

57'6 

3'25 

3-26 

3*06 

j» 

63*2 

3'57               3-59 

3-36 

70*0 

3*85               3*87 

3-79 

»»  . 

76*6 
86*0 
137-0 
235-0 
331'0 

4-10 
4-40 
5*26 
570 

4-I3 

4'43 

5-8.6 
6-06 

4-36 

pi. 

6-16 

j> 
Koref, 
Magnus. 

416-0 

5-87 

6*21 

6*36 

11 

550-0 

6*36 

6-54 

" 

1  These  two  values  were  based  upon  erroneous  measurements  of 
Rubens  and  Hollnagel,  but  the  actual  error  introduced  is  small.  The 
same  remark  applies  to  NaCl  and  KBr. 


520       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 
NaCl.    j80v  is  the  mean  of  the  two  values  265*2  and  309*3. 


T  absolute. 

Cv  calculated. 

C    calculated. 

.  C    observed,    j 

Observer. 

25*0 

0*32 

0*32 

0*29 

Nernst. 

25'5                o*34 
28*0                0-48 

0'34 
0-48 

0-3I 
0*40 

» 

67'5                2*87 

2*88 

3*06 

» 

69*0                2*94                2*95 

3^3 

j  > 

8i'4                3'47                3*49 

3'54 

,, 

83-4                3*6  1                 3*64 

375 

55 

138*0                4*82                4-90 

4*87 

Koref. 

235*0 

5'52                     573 

576 

y 

KBr.  j80v  is  the  mean  of  the  two  values  168*6  and  194*8  (Cjp  was  calcu- 
lated from  Cv  by  the  usual  formula,  the  value  of  A  being  taken  as 
identical  with  that  for  NaCl). 


T  absolute. 

Cv  calculated. 

!  Qp  calculated. 

CL  observed. 

Observer. 

787 

4-67 

!        0° 

474 

Nernst. 

82-5 

477 

4-80 

476 

}  j 

85-4                4-82 

4-85 

4-82 

89-2 

4-91 

4'  94 

5-03 

137-0 

5  '47 

5-42 

Koref. 

234-0 

579 

6*02 

6'io 

»  » 

The  agreement  between  calculated  and  observed  values  is 
seen  to  be  extremely  good.  The  agreement  found  in  the  case 
of  the  salts  is  particularly  interesting  since  the  value  of  v  was 
obtained  in  these  cases  by  optical  means,  t.e.  the  Reststrahlen. 
It  shows  in  a  very  decisive  way  that  the  optical  (infra-red) 
vibration  and  thermal  vibration  of  the  atom^  which  gives  rise  to 
internal  energy  (the  atom  being  supposed  electrically  charged, 
of  course,  since  light  is  an  electromagnetic  phenomenon),  are 
identical.  (The  values  of  v  for  the  metals  was  obtained  by  the 
aid  of  Lindemann's  melting-point  formula.) 

We  may  conclude  our  account  of  the  Nernst-Lindemann 
empirical  expression  by  giving  the  results  obtained  in  the  case 
of  the  diamond  over  the  entire  range  from  30°  abs.  to 
1169°  abs, 


SPECIFIC   (ATOMIC]    HEAT  OF  SOLIDS        521 


ATOMIC  HEAT  OF  DIAMOND. 

0v»  =  1940  (obtained  by  using  Lindemann's  formula,  the  melting-point 
being  taken  as  3600°). 


T  absolute. 

Cv  calculated. 

C    calculated. 

C    observed. 

Observer. 

30 

O'OOO 

O'OOO 

O'OO 

Nernst. 

42 

O'OOO 

O'OOO 

O'OO 

88 

0-006 

o-oo6 

0-03 

92 

0-009 

0-009 

0-03 

205 

0-62 

O-62 

0-62 

209 

0-65 

0-65 

0-66 

220 

0-74 

0-74 

0-72 

222 

0-78 

0-78 

0-76 

Weber. 

232 

0-87 

0-87 

0-86 

Koref. 

243 

0-97 

0-97 

0'95 

Dewar. 

262 

ri6 

1-16 

1-14 

Weber. 

284 

i'37 

i;37 

I'll 

306 

i  '59 

33i 

1-82 

1-83 

1-84 

358 

2-07 

2-08 

2'12 

413 

2'53 

2'55 

2-66 

1169 

5''9 

5'4i 

5'45 

Again  the  agreement  between  theoretical  and  observed 
values  is  good.  The  important  point  about  the  diamond,  as 
already  remarked,  is  that  it  ceases  to  have  any  measureable 
heat  capacity  at  all  for  about  40°  above  absolute  zero.  This 
is  in  agreement  both  with  the  thermodynamic  theorem  of 
Nernst  and  the  Einstein  theory  of  specific  heat. 

As  regards  the  theoretical  significance  of  the  Nernst- 
Lindemann  modification  of  the  Einstein  expression  for  atomic 
heat  practically  nothing  can  be  said  (see,  however,  Nernst  and 
Lindemann,  I.e.). 

Reference  may  also  be  made  at  this  point  to  the  fact  that 
attempts  have  been  made  to  treat  the  energy  of  polyatomic 
gases  from  the  standpoint  of  the  unitary  theory  of  energy  as 
applied  to  vibration  and  rotation.  An  account  of  this  will  be 
found  in  the  papers  of  Nernst  and  Lindemann  already  referred 
to,  as  well  as  in  a  paper  by  N.  Bjerrum  (Zdtsch.  Elektrochem.t 
17,  731,  1911). 


522       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

The   Specific  Heat  of  Solids  at  High    Temperatures   and  the 
Significance  of  the  Energy  possessed  by  the  Electrons. 

It  will  be  remembered  that  the  expression  for  the  atomic 
Iheat  (at  contant  volume)  of  a  solid,  whether  stated  in  the 
•original  form  of  Einstein  or  in  the  Nernst-Lindemann 
•modification,  reaches  as  a  limit  the  value  3R  (5*95  cals.).  As 
;a  matter  of  fact,  however,  the  atomic  heat  (at  constant  volume] 
ihas  been  found  to  exceed  this  limit  considerably  when  the 
'temperature  to  which  the  value  of  the  atomic  heat  refers  is 
very  high.  At  these  temperatures  there  must  be  some  other 
source  of  internal  energy  content  not  taken  account  of  in 
simply  ascribing  the  total  energy  to  the  vibrations  of  the  atoms. 
At  high  temperatures  the  body  becomes  "  white  hot,"  that  is 
it  is  capable  of  emitting  short  waves  in  the  visible^  which  are 
due  in  all  probability  (as  we  have  already  seen)  to  the  vibra- 
tions of  the  electrons  in  the  atoms.  Although  the  energy  of 
the  electrons  can  be  neglected  at  low  temperatures  compared 
to  that  possessed  by  the  atoms,  this  is  no  longer  the  case  at 
high  temperatures,  and  hence  such  must  be  taken  into  account 
in-  any  expression  for  specific  (or  atomic)  heat.  An  attempt 
in  this  direction  has  been  made  by  J.  Koenigsberger  (Zeitsch. 
Elektrochem.)  17,  289,  1911),  who  considers  that  the  atomic 
heat  of  metals  (at  constant  volume)  should  reach  9  instead 
of  6,  by  taking  the  vibrating  energy  of  the  electrons  into 
account.  No  great  stress,  however,  can  be  laid  upon  the 
significance  of  the  actual  numerical  limit  quoted  (cf.  Nernst 
and  Lindemann,  I.e.).  It  is  interesting,  however,  to  see  the 
numerical  values  for  the  atomic  heat  Cw  actually  obtained  at 
high  temperatures  (cf.  Koenigsberger,  Lc.}  \ — 

ATOMIC  HEAT  (AT  CONSTANT  VOLUME)  AT  HIGH  TEMPERATURES. 


Metal. 

Region  of  temperature. 

"' 

Observer. 

Silver    . 

907-1  100°  C. 

6-7 

Pionchon  (1887). 

Tin  .      .      . 

1100° 

9-2 

•  j 

Iron 

1200° 

9-6 

„ 

Nickel  .      . 

1  150° 

7'9 

it 

Copper 
Aluminium 

900° 
650° 

77 

Richards  (1893). 
Boutchew. 

INFRA-RED  AND    ULTRA-VIOLET   VIBRATION     523 

In  connection  with  the  question  of  the  behaviour  of  the 
electrons  in  solid  conductors  (at  ordinary  temperatures)  some 
very  interesting  work  has  recently  been  done  by  Pringsheim 
{Verhl.  d.  deittsch.  physik.  Gesell.,  1910)  on  the  so-called  "selec- 
tive photo-electric  effect."  Pringsheim  observed  that  visible 
or  ultra-violet  light  polarised  in  a  certain  plane  impinging  on 
metallic  surfaces  caused  an  emission  of  electrons.  The 
frequency  of  the  incident  light  had  to  be  confined  to  a  fairly 
narrow  range,  which  varied  from  metal  to  metal,  in  order  that 
electrons  might  be  emitted.  The  electrons  present  in  a 
metal  seem,  therefore,  to  possess  a  characteristic  vibration 
which  we  can  denote  by  Wioiet,  and  when  light  of  the  same 
frequency  strikes  the  metal  the  electrons  may  be  caused  to  so 
far  increase  their  amplitude  that  some  are  ejected  from  the 
metal  altogether.  An  expression  for  this  electron  vibration 
frequency  has  been  deduced  by  Lindemann  ( Verhl.  d.  deutsch^ 
physik.  GeselL,  13,  482,  1911)?  which  reproduces  the  experi- 
mental values  exceedingly  well.  To  indicate  the  region  of 
the  spectrum  to  which  this  selective  effect  belongs,  it  may  be 
mentioned  that  for  rubidium  the  characteristic  A  is  480/411; 
for  potassium,  440;^ ;  for  sodium,  320/416;  and  for  barium, 
280/4^.  Haber  (Verhl.  d.  deutsch.  physik.  Gesell.^  13,  1117, 
1911)  has  taken  up  this  problem  and  has  shown  a  very  simple 
and  interesting  relation  between  the  "iMoiet"  and  the 
Einstein  infra-red  characteristic  frequency  of  the  atoms  of  the 
same  substance,  which  we  may  denote  by  "  Vmfra-red."  This 
relation  is  simply — 

Vviolet    /M 

^infra-red        ^    ^ 

where  M  is  the  mass  of  the  atom  and  ;;/  the  mass  of  an 
electron.  Haber  has  shown  that  this  agrees  extremely  well 
with  observed  values.  Thus,  using  Lindemann's  formula  just 
referred  to,  to  calculate  Vvioiet,  one  can  make  use  of  the  Haber 
relation  to  calculate  ^infra-red  and  compare  it  with  the  values 
given  by  Lindemann's  melting-point  formula  (for  calculating 
^infra-red).  The  results  are  as  follows — 


524       A   SYSTEM  OF  PHYSICAL    CHEMISTRY 


Substance.  '       Vviolet  X  io-lr'. 


infra-red 


"infra-red  X  «** 
calculated  by  Linde- 


Li 

1-27 

11-26 

Na 

0-947 

4-62 

K 

0-685 

2*55 

Rb 

O'6l2 

I-55 

Cs 

0-546 

1-108 

I  . 

0-906 

* 

calculated  by  Haber.     mann's  melting-point 
formula. 


1078 
4-38 
2'57 
I-56 

i'i3 
1-85 


THE  CONNECTION  BETWEEN  THE  PLANCK-EINSTEIN  THEORY 
OF  THE  SPECIFIC  (ATOMIC)  HEAT  OF  SOLIDS  AND  THE 
THERMODYNAMICAL  (HEAT)  THEOREM  OF  NERNST. 

We  have  already  seen  that  at  very  low  temperatures  the 
unitary  theory  of  energy  leads  one  to  expect  that  the  atomic 
heat  (necessarily  of  solids  and  supercooled  liquids)  becomes 
practically  zero.  That  is,  Einstein's  formula  predicts  that  — 


limit 


=  O 


nr/r=o 

This  is  in  agreement  with  Nernst's  heat  theorem,  which  may 
be  stated  thus — 

limit  T  =  o.         —  =  o 

limit  T  =  o         —  =  o 

To  see  how  the  latter  relation  regarding  free  energy  (A)  is 
also  in  agreement  with  the  kinetic  hypothesis  of  Einstein,  we 
may  proceed  as  follows :  As  the  molecules  of  a  solid  at  very 
low  temperatures  do  not  possess  any  sensible  kinetic  energy, 
their  mutual  distance  apart  will  change  very  little  with  the 
temperature.  From  this  it  follows  that  their  mutual  potential 
energy  will  also  remain  practically  unchanged,  and  hence  their 
free  or  available  energy  (which  depends  upon  the  potential 
energy)  will  also  remain  unchanged.  That  is  in  the  limit  when 

T  =  o,  -FP=O.     We   can   reach   the   same   conclusion   in  a 
a  \ 


NERNSrS  HEAT  THEOREM        525 

slightly  different  way.  We  know  that  the  energy  of  radiation 
and  therefore  the  amplitude  of  the  resonators  (i.e.  the  atoms) 
increases  with  the  temperature  according  to  a  power  law 
considerably  greater  than  the  first.  Hence  as  temperature 
falls  the  amplitude  will  decrease  much  more  rapidly,  finally 
becoming  exceedingly  small.  That  is  again  the  potential 
energy  (and  therefore  the  free  energy)  may  be  regarded  as 
independent  of  temperature  when  the  temperature  is  low.  It 
cannot  be  said,  of  course,  that  the  validity  of  Einstein's  theory 
is  a  "  proof"  of  the  validity  of  Nernst's  heat  theorem.  They 
are  in  agreement,  however,  as  regards  fundamental  assump- 
tions. For  a  further  discussion  of  the  relation  between  the 
two,  the  reader  is  referred  to  :  Nernst  (Zeitsch.  Elektrochem., 
17,  273,  1911);  Sackur  (AnnaL  Physik.,  [4]  34,  455,  1911); 
F.  Juttner  (Zeitsch,  Elektrochem.,  17,  139,  1911). 

It  may  be  pointed  out  that  at  absolute  zero  of  temperature 
on  Einstein's  theory,  the  atoms  possess  no  vibrational  energy 
at  all.  If,  however,  we  are  to  distinguish  an  atom  of  one 
kind  of  substance  from  another,  we  must  suppose  that  each 
possesses  a  specific,  intrinsic,  or  internal  energy  independent  of 
the  temperature.  This  quantity  is  denoted  by  U0  in  Nernst's 
nomenclature. 


CALCULATION  OF  AFFINITY  FROM  THERMAL  DATA  BY  THE 
SIMULTANEOUS  APPLICATION  OF  NERNST'S  HEAT  THEOREM 
AND  EINSTEIN'S  THEORY  OF  ATOMIC  HEAT  (IN  THE  FORM 
OF  THE  NERNST-LINDEMANN  EQUATION). 

For  the  sake  of  simplicity,  we  shall  consider  an  actual  case, 
namely,  the  affinity  of  silver  for  iodine,  recently  investigated 
from  this  point  of  view  by  Ulrich  Fischer  in  Nernst's  labora- 
tory (Zeitsch.  Elektrochem.,  18,  283,  1912). 

The  first  question  taken  up  by  Fischer  is  the  heat  of  the 
reaction — 

2Ag     +     I2     =     2AgI 

(solid)  (solid)  (solid) 

The  direct  measurements  of  Thomsen  and  Berthelot  being 


526       A    SYSTEM    OF  PHYSICAL   CHEMISTRY 

unsatisfactory,    recourse   was    had    to    tlie    Gibbs-Helmholtz 
equation  — 


:as  applied  to  the  measurement  of  the  e.m.f.  of  the  cell — 
Ag  |  KI  solution,  saturated  with  Agl  |  I2 

at  several  temperatures. 

The  results  were  as  follows  : — 

N 
Employing      KI  solution,  Evoits  =  0*6948  -f-  o-ooo305/ 

2 

N 
„  —  „  Eyolts  =  0-693 2  4- 0-000305^ 

N 
„  ,,  Eyoits  =  0*6916 -)- o'ooo3o5/ 


the  temperature  t  being  given  in  degrees  centigrade. 

These  values  for  the  e.m.f.  require  a  correction,  however, 
owing  to  the  fact  that  some  of  the  iodine  from  the  right-hand 
electrode  dissolves  in  KI,  giving  tri-iodide  (and  other  poly- 
iodides,  the  quantity  of  which  may,  however,  be  neglected). 
This  causes  the  concentration  of  the  iodine  ion  I',  on  the 
iodine  side  of  the  cell,  to  be  diminished,  and  the  concentration 
difference  of  I'  on  the  two  sides  of  the  cell  give  rise  to  an 
e.m.f.  e,  where — 

e  =  RT  loge  C± 

~  being  the  ratio  of  the  I'  ion  concentrations.     With  the  help 

of  the  data  of  von  Ohlendorff,  and  some  experiments  of  his 
own  on  the  solubility  of  iodine  in  potassium  iodide,  Fischer 
succeeded  in  calculating  the  value  of  e  for  the  three  cases  con- 

N    N  N 

sidered,  viz.  — »  —  ,  and  —  KI  respectively.      The  following 
23  10 


tiERNST'S  HEAT  THEOREM  5-27- 

table   contains   the   corrected  value    of  the   e:m.C,  and   the. 
temperature  coefficient  -      :— 


KI  solution.        Temperature, 

e. 

E  corrected.                 —  — 

-                  I          23  '-8 
'  I         35-3 

0-01869 

0'68l7            J 
0-6833             -      2*419 
0-6839            J 

*-     .        .       .    J!            24-6 

5                  I         38-6 

0-01719 
0-01938 
0*02327 

0-6805            1 
0-6813                   I'267 
0-6817            ) 

[H  i 

0-01658 
0-01848 
0-02II8 

0-6791           |j 

0-6805         J     1*66 
0-6807         ) 

Employing  these  data  in  the  Gibbs-Helmholtz  equation  > 
Fischer  found  for  15°  C.  (288°  abs.)  :— 

N 

KI  solution,  U  =  0*6821  X  23046  —  288  X  2*419 


N 


N 


10 


=  15023  cals. 
KI  „  U —  0-6806  X  23046  —  288  X  i '267 

=  15315  cals. 
KI  „  U  — 0-6791  X  23046  —  288  X  i'66 

=  15216  cals. 
The  mean  value  of  U  is  15169  cals. 

Besides  this  indirect  method  of  measuring  U,  Fischer  also 
employed  a  calorimetric  method,  precipitation  of  insoluble  salt 
being  avoided  by  the  following  very  ingenious  procedure  due 
to  von  Wartenberg.  As  solvent  a  3N  KCN  solution  was 
employed,  and  in  this  was  placed  a  known  weight  of  silver 
powder.  The  silver  is  not  attacked  by  the  KCN.  To  the 
liquid  was  added  a  known  weight  of  iodine,  the  following 
reaction  taking  place — 

Ag  +  I  +  2KCN  =  AgK(CN)2  +|KI  .  .  .  .  (i> 


528       A   SYSTEM  OF  PHYSICAL    CHEMISTRY 

The  heat  effect  of  this  reaction  was  measured.  Then 
using  a  similar  solution  of  KCN,  a  known  weight  of  silver 
iodide  was  added  (no  silver  powder  being  present  in  this  case), 
the  resulting  reaction  being — 

Agl  +  2KCN  =  AgK(GN)2  +  KI    .     .     (2) 

The  difference  of  the  heat  effects  of  the  reactions  (i)  and 
(2)  gives  the  heat  effect  desired,  viz. — 


The  mean  value  thus  obtained  was  U  =  14280  cals.,  which 
agrees  pretty  closely  with  the  result  of  the  first  method. 
Fischer  made  use  of  still  another  calorimeter  method,  in 
which  some  of  Thomsen's  data  on  other  salts  were  employed, 
the  results  being  less  trustworthy.  [Note  that  this  is  a  heat 
effect  at  constant  volume,  since  the  system  does  not  alter 
appreciably  in  volume.] 

Now,  if  we  consider  the  cell — 

Ag         Agl     |     I2 

(solid)          (solid)         (solid) 

this  represents  a  condensed  system,  each  phase  being  a  single 
substance  (no  solution  or  vapour  being  present),  and  we  can 
apply  Nernst's  heat  equations  directly,  viz. — 

U  =  U0  +  aT  +  /3T2 
and  A  =  U0  —  aT  —  °  T2 

[Note  that  the  "  A  "  or  "  E  "  of  this  cell  is  not  quite  the 
same  as  A'  for  the  cell  in  which  the  KI  solution  is  present. 
The  heat  effect  in  both  cases  is,  however,  the  same,  for  as 
we  have  seen  the  heat  effect  as  calculated,  say  by  the  first 
method,  is  independent  of  the  KI  concentration,  being  simply 
the  heat  effect  of  the  reaction  Ag  +  I  ->  Agl.]  If  we  knew  by 
experiment  the  specific  or  atomic  heats  of  the  substances  Ag, 
I2,  and  Agl,  over  a  range  of  temperatures,  we  could  calculate 
the  coefficients  a  and  j8,  as  we  have  already  done  in  previous 
illustrations  of  the  Nernst  theorem.  But  instead  of  proceeding 
thus,  we  can  apply  the  Nernst-Lindemann  expression  for  these 


HEAT   THEOREM 


529 


atomic  heats  (since  the  substances  are  all  solids),  which  as  we 
have  seen  requires  simply  a  measurement  of  the  characteristic 
vibration  frequency  v  of  each  substance.  Suppose  such  fre- 
quencies are  known  (determined  by  any  of  the  three  methods 
already  given),  we  can  calculate  separately  the  specific  or 
atomic  heats  of  the  reactants  (silver  and  iodine),  and  the 
resultant  Agl.  We  then  make  use  of  the  relation  — 


heat  capacity)  _  (heat  capacity)  __  ^U 
of  reactants    j      (of  resultants  )  ~~  ^T 


_     fi 

"  2^ 


In  general  the  y  term  may  be  neglected  compared  to  the  j8  term, 
and  hence  )3  is  obtained  at  once.  Fischer,  as  a  matter  of 
fact,  makes  the  substitution  of  the  Nernst-Lindemann  expression 
directly  in  the  thermodynamic  equations  of  Nernst. 

By  making  use  of  the  values  of  PQV  and  a  obtained  in  this 
manner,  Fischer  calculated  U  for  a  temperature  T  =  288, 
finding  the  value  15079  calories,  a  quantity  which  agrees 
extremely  well  with  the  former  values  which  were  obtained 
for  U  quite  independently  of  the  Nernst  heat  theorem,  or  the 
Nernst-Lindemann  atomic  heat  expression.  It  is  to  be  re- 
membered that  the  U  and  A  terms  here  refer  to  the  reaction 
between  pure  solid  Ag  and  solid  I2,  giving  solid  Agl.  A  is 
therefore  the  affinity  which  we  set  out  to  calculate.  The 
following  table  contains  the  values  of  A  and  U  thus  obtained  — 


AFFINITY  OF  SILVER  FOR  IODINE  AT  VARIOUS  TEMPERATURES. 


Temperature, 
degrees  absolute. 

U  in  calories  per  gram- 
atom  of  silver. 

A  in  calories  per  gram- 
atom  of  silver. 

0 

15,166 

15,166 

20 

*5>*53 

15,  173 

40 

15,136 

15,201 

60 

15,124                      15,235 

80 

100 

1  5!  1  14 

I5,3H 

I40 

15,108 

15,394 

1  80 

15,101 

15,477 

22O 

15,093                      15,5^0 

260 

15,085 

15,650 

300 

15,074 

15,733 

T.P.C. — II. 


2  M 


530       A    SYSTEM   OF  PHYSICAL    CHEMISTRY 

The  accompanying  figure  (Fig.  98)  shows  diagrammatically 
the  relative  variation  of  U  and  A.  It  will  be  observed  that, 
in  this  reaction,  the  affinity  increases  with  increase  in  tempera- 
ture, whilst  the  heat  effect  or  total  energy  change  decreases. 


U 


100 


300°T 


FIG.  98. 


A.  Magnus  (Zeitsch.   Elektrochem.^  16,  273,   1910)  has  carried 
out  a  similar  investigation  of  the  reaction — 

Pb  +  2AgCl-»  PbCl2  +  2Ag 

in  which  the  affinity  was  found  to  decrease  as  the  temperatui 
was  raised. 


DEBYE'S  EQUATION  FOR  THE  ATOMIC  HEAT  OF  SOLIDS. 
(P.  Debye,  Ann.  Physik.,  [4]  39,  789,  1912.) 

We  have  seen  that  Einstein's  specific  (atomic)  heat  formula 
only  holds  qualitatively  and  that  whilst  that  of  Nernst  and 
Lindemann  accords  much  better  with  experiment,  their  ex- 
pression suffers  from  the  very  serious  drawback  of  lacking 
a  sound  theoretical  basis.  Debye  has  recently  developed  an 
expression  having  a  theoretical  basis — employing  the  quan- 
tum hypothesis — and  has  found  that  the  expression  repro- 
duces experimental  values  with  even  greater  fidelity  than 


DE BYE'S  EQUATION  531 

that  of  Nernst  and  Lindemann.  Debye's  formula  may  there- 
fore be  regarded  as  the  most  significant  of  all  specific  heat 
formulae  yet  proposed.  The  general  line  of  thought  pursued 
by  Debye  is  as  follows  l : — 

In  Einstein's  investigation  it  was  assumed  for  the  sake  of 
simplicity  that  the  vibrating  particle  (the  atom)  only  gave  rise 
to  monochromatic  radiation  (frequency  v)  and  only  absorbed 
such.  Instead  of  this  limitation  to  a  single  characteristic 
frequency  v,  Debye  works  out  the  expression  for  atomic  heat 
on  the  basis  of  absorption  (and  emission)  of  a  number  of 
vibration  frequencies,  in  fact  a  whole  spectrum.  It  will  be 
seen  at  once  that  this  is  far  more  likely  to  coincide  with  what 
actually  occurs  than  the  simpler  assumption  of  Einstein. 
Debye  starts  out  with  the  very  plausible  assumption  that  a 
vibrating  atom  in  a  solid  cannot  be  vibrating  simply  harmoni- 
cally with  a  single  frequency,  but  owing  to  the  proximity  of 
other  atoms  and  probably  to  collisions  with  them  the  mode 
of  vibration  will  be  complex.  Such  complex  modes  of 
vibration  can  be  treated,  as  Fourier  showed,  as  theoretically 
made  up  of  a  series  of  true  simple  harmonic  motions,  and  we 
have  thus  to  integrate  over  a  spectrum  of  frequencies  if  we 
wish  to  calculate  with  greater  exactness  the  total  energy 
content  of  a  vibrating  atom.  The  first  point  to  be  noted  is, 
however,  that  we  do  not  deal  with  a  spectrum  extending  from 
i/  =  o  to  v  =  oo  .  If  a  body  consists  of  N  atoms — treated  as 
massive  points — the  system  possesses  3N  degrees  of  freedom. 
The  system  will  therefore  in  general  exercise  3N  different 
periodic  vibrations,  i.e.  3N  different  vibration  frequencies. 
If  the  older  view — consonant  with  the  principle  of  egui- 
partition  of  energy — were  true,  namely,  that  the  energy  was 
the  same  for  each  degree  of  freedom  over  the  entire  spectrum, 
then  we  could  state  directly  that  each  vibration  frequency 

T> 

corresponded  to  the  energy  £T  (where  k  =  — ,  N  being  now 

the  number  of  atoms  in  i  gram-atom)  so  that  the  total  energy 
of  the  body  would  be  simply  3NRT  per  gram-atom  as  Dulong 
and  Petit's  Law  requires. 

1  For  details  see  Debye's  paper. 


532       A    SYSTEM    OF  PHYSICAL    CHEMISTRY 

We  have  seen,  however,  that  the  whole  point  of  the 
quantum  theory  is  the  negation  of  the  principle  of  equi- 
partition  throughout  a  spectrum,  and  that  instead  the  energy 
per  vibration  varies  with  the  type  of  vibration.  This  lack  of 
equipartition  is  expressed  as  we  have  already  seen  in  the 

hv 
Planck  expression  ^v/JeT  __--,  which  gives  the  true  mean  energy 

of  any  single  vibration  v.1  To  obtain  the  total  energy  of  the 
vibrating  system  it  is  necessary  to  sum  this  expression  over 
the  spectrum  (of  absorption  or  emission),  the  spectrum  not 
containing  an  infinite  number  of  frequencies  but  limited  to  3N, 
as  already  pointed  out.  The  spectrum  is  characterised  by  two 
factors,  (a)  its  boundaries,  (b)  the  density  of  the  lines,  i.e.  the 
number  of  lines  in  any  given  vibration  region  dv.  There  is, 
according  to  Debye,  a  certain  definite  limiting  frequency  vtn, 
beyond  which  the  spectrum  does  not  extend.  Debye  has 
reached  a  number  of  important  conclusions  in  the  course  of 
his  investigation.  The  first  is  this :  If  the  temperature  T  be 
regarded  as  a  multiple  (or  stibmultiple)  of  a  temperature  6  (a 
characteristic  constant  for  any  given  substance) ,  then  the  atomic 
heat  for  all  monatomic  substances  can  be  represented  by  the  same 

T 

curve,  i.e.  the  atomic  heat  is  a  universal  function  of  z 

u 

A  second  relation  which  is  not  confined  to  monatomic 
substances  states  that  the  number  of  lines  spread  over  a  region 
dv  is  proportional  to  v^dv  (a  relation  for  black-body  radiation 
already  obtained  by  Jeans).  From  this  Debye  concludes  that 
at  siifficiently  loiv  temperatures  the  atomic  heat  of  all  solids  is 
proportional  to  T3,  that  is  to  the  third  power  of  the  abolute 
temperature.  This  conclusion  differs  (theoretically)  widely 
from  the  conclusion  of  the  Einstein  and  the  Nernst-Lindemann 
equations.  It  is  borne  out  by  the  experimental  data,  but 
experiment  at  low  temperature  bears  out  the  Nernst-Linde- 
mann view  equally.  As  a  corollary  it  follows  that  the  atomic 
energy  content  at  low  temperatures  is  proportional  to  the  fourth 
power  of  the  absolute  temperature  (cf.  Stefan's  Law  for  total 

1  Debye,  it  may  be  mentioned,  takes  as  the  most  reliable  numerical 
value  for  h,  7*10  X  icr27  erg  sees. 


DEBY&S  EQUATION  533 

radiation).     Debye's  expression  for  the  energy  IT  of  a  solid 
of  volume  V  which  contains  N  atoms  is — 


the  energy  not  being  ascribed  to  a  single  harmonic  vibration 
i>,  but  to  a  spectrum  of  such  extending  from  o  to  vm.  "  If  we 
now  define  the  temperature  6  (the  characteristic  constant  of 
the  substance)  by  the  expression  — 


(a) 


and   introduce   into   equation   (i)   as   a    new    dimensionless 
variable  —  the  magnitude  — 

«.       hv 


we  can  write — 


MJrt/*T\»f      &d£ 

U  =  9N£T(y-  ) 

\/ivm/  £  —  i 

*  o 

or,  substituting  6  for  -—-,  we  obtain — 

K>  \. 


...-..-  <« 

Dulong  and  Petit's  law,  if  true,  would  require  the  relation  — 


U  = 


n 

Now  if  we  write  —  =  x,  and  differentiate  equation  (5)  with 
respect  to  temperature,  we  obtain  finally  — 


The  magnitude  3NR  has  the  value  5-955  cals.  and  may  be 
denoted  by  C^  since  this  numerical  limit  is  reached  at  high 


534       A    SYSTEM  OF  PHYSICAL    CHEMISTRY 

temperatures.     Equation  (6)  may  therefore  be  rewritten  in  the 
form — 

C*          12   I        £3^£  30: 


„!  *_,    ,._,    •••(?) 

J   0 

Equation  (6)  or  (7)  is  Debye's  atomic  heat  formula. 

For  the  case  in  which  x  is  small,  i.e.  T  is  large  (compared 

to  0),  -j—  -  becomes  £2,  and  x__     becomes  —^  or  simply  3. 
Hence, ~-  =       —  3  =  i,  in  other  words  Dulong  and  Petit's 

^^oc  O 

Law  holds  at  high  temperatures.     On  the  other  hand,  when 
x  is  large,  i.e.  T  is  small,  we  obtain  by  expanding 
the  series  ^(e~^  -j-  e~^  +  e~3*  +  •  •  0^>  or — 


f  m 

T=^ 

J   0 


Hence  equation  (7)  yields  —  since  the  series  expression  in  the 
bracket  is  1-0823  — 

C,;       12X6x1-0823 


That  is,  at  sufficiently  low  temperatures,  the  atomic  heat  is 
proportional  to  the  cube  of  the  absolute  temperature  T.    (The 

above   relation   can  be   used  to  calculate  6,  employing  the 

p 
experimental  value  of  fr~  at  some  low  temperature.) 

^oo 

Debye  has  tested  his  formula  (equation  (6)  or  (7))  and  found 
it  to  agree  very  well  with  experiment.  It  has  further  been 
tested  independently  by  Nernst  and  Lindemann  (Sitzungsber. 
BerL  Akad.,  p.  1160,  1912)  with  the  following  results,  in 
which  the  values  of  Q,  (atomic  heat  at  constant  pressure)  are 
obtained  from  the  values  of  C«  given  by  Debye's  formula  by 
employing  one  of  the  relations  between  Cp  and  Cv  mentioned 
at  an  earlier  stage.  Nernst  and  Lindemann  slightly  alter  the 
form  of  Debye's  equation  (in  respect  of  the  symbols)  so  as  to 


DEBY&S  EQUATION  535 

bring  it  more  into  line  with  previous  expressions.     The   ex- 
pression as  used  by  them  takes  the  form — 


ALUMINIUM  (0  =  fiv  =  398  ;  earlier  value  employed  —  405). 


T. 


86-0 
88-3 
137-0 
235-0 
33i-o 
433 '° 
555!o 


Cp  observed. 

C    calculated. 

Difference 
obs.-calc. 

Difference  obs.-calc. 
(formula  of  Nernst 
and  Lindemann). 

0-25 

0-25 

O'OO 

+  0-02 

°'33 

0-32 

+  0-01 

+  0-02 

2-41 

2-36 

+  0-05 

~0'02 

2-52 

2-50 

+  0-02 

—  O'OI 

2-62 

2*59 

+  0-03 

O'OO 

3'97 

4*  10 

-0-13 

-0'08 

5'32 

5*34 

-0'02 

+  0-02 

5-82 

5*78 

+  0-04 

+  0'06 

6'io 

6-07 

+  0-03 

+  0-04 

6-48 

6-30 

+  0-18 

+  0-18 

DIAMOND  (fiv  =  1860  ;  earlier  value  1940). 


T. 

C    observed. 

C     calculated. 

Difference 
obs.-calc. 

Difference  obs.-calc. 
(formula  of  Nernst 
and  Lindemann). 

88 

0-028 

0-049 

-0'02I 

+  0'022 

92            0-033 

0-058 

-0-025 

+  0-024 

205 

0-618 

0-61 

+  0-008 

o-oo 

209 

0-662 

0-66 

+  0-002 

+  Q-OI 

220 

O'722 

o'74 

—  O'OlS 

—  0-64 

222 

0-76 

0'75 

-f  O'OI 

—  O"O2 

243 

0'95 

0-925 

+  0-025 

—  O'O2 

262 

I-I4 

I  '10 

+  0-04 

—  0-02 

284 

I'35 

i  -32 

+  0-03 

—  O'O2 

306 
331 

I-58 
I-84 

i*S4 

1-82 

+  0-04 
+  0'02 

-0  01 
+  O'OI 

358                 2-12 

2-07               -f-o'o:; 

+  0-04 

413 

2-66 

2'6l 

+  0-05                  +0-II 

1169 

5'45 

5-49            -0-04 

-f  o  04 

536       A    SYSTEM   OF  PHYSICAL   CHEMISTRY 

Silver  and  copper  give  equally  good  results,  as  also  do 
potassium  chloride  and  sodium  chloride. 

NOTE  on  the  method  of  evaluating  the  summation  term  of  the  Debye 
formula  in  the  form  given  by  Nernst  and  Lindemann. 
To  evaluate  — 


Setf~J  equal  to  A:     Also  set  c  l  =  a.     Then  e~nx  =  anx  =  (a*)11  -  yn  if  y 

be  set  equal  to  ax. 

The  sum  can  therefore  be  written  : 


ttx      n"x" 


or  the  same  expression  may  be  written  : 


»  =  1 


First  term.  Second  term. 


r2      j'3  1 

First  term  =  -\-+  .   +< ^   +  .  .  .  ^/ ////    . 


Second  term  = 


Third  term. 

y3 

3 


Fourth  term. 


Third  term  =  ^  f^  +^  +  ^  +-^  +  ...  ad  inf\ 

term  -  6,  [^  4-^j  +J'l  +y\  +  '  '  '  ^  '«/•]. 
•*"  L1       2       3       4  J 


Fourtli 


Each  of  these  series  is  convergent,  so  that  only  the  first  three  or  four 
expressions  in  each  need  be  taken  into  account. 


EINSTEIN*S  PHOTOCHEMICAL   LAW          537 

Einstein's  Photochemical  Law >  or  the  "Law  of  the  Photochemical 
Equivalent" 

(A.  Einstein,  Ann.  d.  Physik.,  [4]  37,  832,  1912.) 

According  to  Einstein,  a  photochemical  reaction  takes 
place  owing  to  the  absorption  of  radiation  in  terms  of  quanta, 
each  single  molecule  of  a  photo-sensitive  substance  requiring  just 
one  quantum  hvQ  (of  the  requisite  frequency  VQ  which  the  substance 
itself  can  absorb)  in  order  that  it  may  be  decomposed.  So  far  as 
experiment  has  gone,  the  law  has  been  approximately  verified. 
Bodenstein  (Zeitsch.  physik.  C/iem.,  85,  329,  1913)  gives  a 
table  of  several  reactions  from  which  it  will  be  seen  that  if  not 
a  single  quantum,  at  least  a  very  small  number,  2  to  5,  are 
required  per  molecule  of  the  substance  photochemically 
decomposed.  Such  discrepancy  as  does  exist  is  probably  to 
be  attributed  to  the  fact  that  the  experimental  conditions  did 
not  correspond  closely  enough  to  those  postulated  by  Einstein, 
the  principal  one  of  which  is  that  there  shall  be  the  same 
intensity  or  density  of  radiation  throughout  the  system.  If  the 
system  absorbs  light  very  strongly,  and  is  only  illuminated  in 
one  direction,  this  condition  will  not  be  fulfilled,  and  instead 
of  reaching  a  true  equilibrium  governed  as  Einstein  supposes 
by  an  expression  analogous  to  the  ordinary  mass  action  law, 
we  shall  meet  with  photochemical  stationary  states  instead, 
such  as  those  realised  in  the  anthracene-dianthracene  reaction. 

Einstein's  general  argument  is  as  follows.  Consider  a  gas 
mixture  containing  3  different  species,  whose  molecular  weights 
are  mlt  w2,  and  MB,  and  suppose  there  are  »lf  «a,  an(^  ;/3  gram- 
molecules  of  each  of  these  present.  The  reaction  considered 
is  one  in  which  i  molecule  of  mt  decomposes  photo-chemi- 
cally  into  i  molecule  each  of  m2  and  ;//3,  an  equilibrium 
point  finally  being  reached.  The  first  assumption  made  is 
that  the  decomposition  of  ;;/]_  proceeds  (owing  to  the  absorp- 
tion of  radiation)  independently  of  the  presence  of  the  other 
species.  The  second  assumption  is  that  the  probability  that 
an  m±  molecule  decomposes  in  a  given  time  is  proportional  to 
the  density  p  of  a  given  monochromatic  radiation  to  which  the 


538       A    SYSTEM  OF  PHYSICAL   CHEMISTRY 

system  is  exposed.  From  these  two  assumptions  it  follows 
that  the  number  Z  of  m^  molecules  decomposing  per  unit 
time  can  be  written — 

where  A  is  a  proportionality  factor  which  only  depends  upon 
the  temperature  T. 

As  regards  the  recombination  process,  Einstein  assumes  it 
to  be  an  ordinary  bimolecular  reaction  (which  emits  the 
radiation  first  absorbed  by  the  m±  molecules).  If  Zj  denotes 
the  number  of  m±  molecules  thus  re-formed  per  unit  of  time,  we 
can  write — 

7  ._  A'   v  ;/2  ;/3 

*"1  /I      .       V      .    — -     •    -IT 

when  V  is  the  volume  of  the  system  ( -^  and  -^  denoting  the 

concentrations  of  each  of  these  species).  A'  likewise  only 
depends  upon  the  temperature,  not  upon  p  (assumption  3), 
and  if  the  temperature  be  kept  constant,  A  and  A'  are 
constant.  When  thermodynamic  equilibrium  is  reached,  i.e. 
equilibrium  between  the  matter  involved  and  the  radiation 
itself,  we  can  equate  the  two  quantities,  supposing  Z  and  Za  to 
refer  to  this  case.  That  is,  we  obtain  the  relation — 

V  'V       A 


Einstein  now  proceeds  to  consider  the  entropy  of  the 
system  undergoing  a  virtual  change,  with  concomitant  change 
in  the  energy  of  the  radiation,  from  which  he  obtains  as  a 
criterion  for  the  above  equilibrium  that — 

p=-'-V*e/KT. 

A 

Where  a  stands  for  a  complex  expression,  involving  the 
temperature  T  of  the  gas  (which  is  independent  of  the  tem- 
perature TV  of  the  radiation),  e  is  the  quantity  of  energy 
absorbed  per  molecule  of  m±  of  the  monochromatic  illumina- 
tion j^0  considered,  whose  density  is  p  and  N  is  the  number  of 


EINSTEIN'S  PHOTOCHEMICAL  LAW  539 
molecules  in  i  gram-molecule.  "  Since  Ts  and  p  are  inde- 
pendent of  the  gas  temperature  T,  the  magnitudes  -r-  and  e 

must  also  be  independent  of  T.  Since  these  quantities  are 
also  independent  of  Ts,  we  arrive  at  the  same  relation  between 
p  and  Ts  as  is  expressed  in  Wien's  radiation  formula."  Since 
Wien's  formula  only  holds  for  the  short  wave  region,  Einstein's 
relation  in  the  first  instance  is  likewise  restricted  to  this  region. 
"  If  we  write  Wien's  radiation  formula  after  introducing  the 
Planck  constants  h  and  £,  we  obtain— 


and  on  comparing  this  with  the  previous  equation,   we  see 
that— 

€  =  hvQ 


K'a 

and  —  -  =  - 

A  <r3 

The  most  important  consequence  of  the  above  is  that 
e  =  /iv0t  which  shows  that  one  gas  molecule  which  decomposes 
under  radiation  of  frequency  VQ  absorbs  in  its  decomposition 
just  hvQ  of  energy,  i.e.  i  quantum,  as  a  mean  value. 

Bodenstein  (loc.  tit.)  has  collected  the  results  of  a  number 
of  investigations  carried  out  by  various  authors,  with  the  object 
of  showing  how  far  the  law  of  the  photo-chemical  equivalent  is 
borne  out  by  experiment.  That  it  is  so  at  least  approximately 
is  clearly  shown  by  the  following  table  :  — 

-n       .•        .    ,-  j  l      Number  of  hv  required  to 

Reaction  studied.  ;  „  decompose  »  One  molecule. 


3O2  =  2O3 i  for  2O3  formed 


2NH3  =  N2  +  3H_ 

2  anthracene  — >  I  dianthracene 


4-5 

Oxidation  of  quinine  by  chromic  ( 
acid     ,  .      .      .      ./ 


2O3  =  3O2  (effected  by  means  oH 
chlorine) j 

Nitro-benzaldehyde  into  nitro- ( 
benzoic  acid / 


4 
07  to  3 


0-8  to  17 


9 


540       A    SYSTEM   OF  PHYSICAL   CHEMISTRY 

Bodenstein  gives  others  (his  so-called  "secondary  light 
reactions),  for  which  the  law  appears  to  break  down  badly, 
one  quantum  decomposing  many  molecules.  All  these  values 
are,  however,  subject  to  very  great  uncertainty. 

In  a  recent  paper  (O.  W.  Richardson,  Phil.  Mag^  March, 
1914),  it  has  been  shown  by  a  different  method  of  deduction 
that  the  Einstein  photochemical  law  may  be  extended  much 
further,  and  is  indeed  quite  general.  That  is  to  say,  even  for 
low  vibration  frequencies  (infra-red)  the  energy  transfer  in  a 
chemical  reaction  will  go  in  terms  of  quanta.  This  seems  to 
open  out  a  new  and  vast  field  for  the  chemical  applications 
of  radiant  energy,  since  apparently  ordinary  or  "  thermal " 
reactions  are  also  in  agreement  with  the  principle  of  the  photo- 
chemical equivalent.  An  application  of  radiation  principles  to 
ordinary  processes  operative  in  solutions  has  already  been  made 
by  F.  Kruger  (Zeitsch.  Elektrochem.,  1911). 


Planck's  Modified  Equation  for  the  Energy  of  an  Oscillator. 
(Cf.  Max  Planck,  Ber.  d.  deut. physik.  Gesell,  13,  138,  1911.) 

The  essential  modification  here  introduced  is  that  while 
emission  of  radiant  energy  takes  place  in  quanta,  and  there- 
fore discontinuously,  absorption  can  take  place  continuously. 
Planck  was  led  to  consider  this  as  more  probable  than  the 
older  view,  one  important  reason  having  to  do  v/ith  the  question 
of  time.  Supposing  absorption  took  place  in  quanta  only, 
then  an  oscillator  might  be  exposed  to  radiation  so  weak  as 
not  to  yield  even  one  quantum,  under  which  condition  the 
oscillator  would  absorb  no  energy  at  all.  This  would  be 
especially  true  for  large  values  of  v  (large  units).  Further, 
even  in  those  cases  in  which  absorption  could  occur,  it  is 
conceivable  that  in  weak  radiation  there  might  be  a  time 
interval. 

On  the  former  view  the  energy  U  of  an  oscillator  could  be 
expressed  by  U  =  ne,  where  n  is  an  integer.  If  absorption 
is  continuous,  the  total  energy  can  no  longer  be  regarded  as 


PLANCICS   .MODIFIED    THEORY  541 

an  exact  multiple  of  e,  but  could  in  general  be  regarded  as 
consisting  of  ne  units  plus  a  quantity  p  over.     That  is — 

U  =  ?i€  +  p 

Now  the  limits  for  the  value  of  p  are  zero  and  one  e.     If 
we  are  considering  a  large  number  of  oscillators  p  will  have 

€       hv 
an  average  value  -  or  —  ,  so  that  we  can  write — 

hv 
U  =  tie  -\ 

2 

On  the  older  view  Planck's  equation  for  U  is — 
U- ^_ 

-  ehv/kT  _  x 

which  evidently  corresponds  to  ne. 

The  new  expression  for  U  can  therefore  be  written — 

U-       —      i  *!!_  =  *?(    _i_       i 

"  ~ 


tf-twri 

Planck  points  out  several  consequences  of  this  new  expres- 

—  hv 

sion.  When  T  =  o,  U  is  not  ==  o,  but  =  — .  This  "  rest- 
energy"  remains  with  the  oscillator  even  at  o°  absolute.  It 
cannot  lose  it  since  it  cannot  emit  anything  less  than  one  hv. 
"  At  high  temperatures  and  for  long  waves  in  the  region 
where  the  Jeans-Rayleigh  Law  holds  the  new  formula  passes 
into  the  old." 

For   short   waves   (visible    and   ultra-violet   region)   *ft»/*T 
becomes  large  compared  to  unity,  and  we  have — 


This  modification  has  considerable  importance  for  photo- 
chemistry. 

As  regards  the  question  of  the  specific  heats  of  solids,  it  is 
pointed  out  by  Planck  that  measurements  of  specific  heat 


542       A    SYSTEM    OF  PHYSICAL    CHEMISTRY 
cannot  be  used  to  compare  the  older  and  newer  forms  of  U, 
since  Ct,   is  -p^,  and  differentiation   removes  the  additional 

hv 
term  —  .     Planck  doubts  if  the  new  expression  can  be  tested 

experimentally  by  any  direct  means.1 

The  survey  which  has  been  given  of  the  Planck-Ein- 
stein quantum  theory  of  specific  heat  will  have  made  it 
clear  that  an  entirely  new  region  of  physico-chemical  inves- 
tigation has  been  opened  up.  For  the  first  time  we  have 
been  able  to  get  at  least  some  idea  of  the  mechanism  of  the 
processes  operative  in  the  solid  state,  which  state  had  for  so 
long  remained  uninvestigated.  It  is  not  unlikely,  of  course, 
that  considerable  modification  will  be  introduced  in  the 
quantum  theory  as  time  goes  on.  The  subject  is  only  in 
its  infancy,  but  there  is  no  doubt  that  the  development  of 
'  physical  chemistry  during  the  next  decade  will  be  along  the 
lines  dealing  with  the  inter-relations  of  radiation  and  matter, 
and  the  various  modes  whereby  radiational  energy  may  be 
transferred  to  and  from  matter,  and  with  this  a  further  insight 
will  be  gained  into  the  real  nature  of  the  property,  which  in 
thermodynamical  language  is  called  "  the  internal  energy  "  of 
a  material  system. 

1  Planck  refers  to  a  paper  by  Stark  (P/iys.  Zeitsch.,  9,  767,  1908),  on 
the  application  of  the  quantum  theory  to  canal  rays. 


INDEX   TO   SUBJECTS 


Absorption  coefficient,  155  seq. 

,  coefficient  of  optical,  411 

,  Henry's  law  of,  150  seq.,  155 

seq. 
Absorptive   power  (for    radiation), 

401  seq. 

Adiabatic  expansion,  26  seq. 
Adsorption,  303  seq. 

equation,  305  seq. 

Affinity,    137,    320   seq.,    354   seq., 

357  seq.,  525  seq. 
and  heat  of  a  reaction,  322  seq. 

electrical    measurement    of, 

339  seq. 

of  carbon   dioxide  for   lime, 

332,  333 

of  hydration,  330  seq.,  368  seq. 

of  hydrogen  for  iodine  vapour, 

328 

for  oxygen,  329,  348 

of  hydrogen  ion  for  hydroxyl 

ion,  345  seq. 

of    oxidation    and    reduction 

processes,  349  seq. 
of  oxygen  for  metals,  333  seq. 

—  of  silver  and  iodine,  525  seq. 
Allotropic  change,  12 1  seq. 

,  dynamic,  295 

Allotropy,  theory  of,  295  seq. 
Amalgam  cell,  194,  195 
Anthracene,     photo-polymerisation 

of,  443  seq. 


Assimilation  of  carbon  dioxide  by 
plants,  429  seq.,  431  seq.,  439  seq. 

Atomic  heat,  473  seq.,  514  seq.,  530 
seq. 

— at  high  temperature,  522 

seq. 


B 

"  Black  body."     See  Full  radiator. 

Boiling  point  and  osmotic  pressure, 

160 

,  effect  of  pressure  on,  46 

Bolometer,  405 
Bound  energy,  129 


Calcium  carbonate  dissociation,  388 

teq. 

Calomel  electrode,  203 
Carbon  dioxide  assimilation  model, 

439  seq. 
"Chemical  constant,"  96  seq.,  380 

seq.,  387 

Chemiluminescence,  408,  409 
Component,  definition  of,  241  seq., 

243 

Concentrated  solutions.  See  Solu- 
tions. 

Concentration  cell  containing  a 
single  solution,  194,  195 


543 


544 


INDEX   TO  SUBJECTS 


Concentration  cell,  llelmholtz's 
mode  of  treatment, 
191  seq. 

,  Nernst's  mode  of  treat- 
ment, 179  seq.,  1 86  seq. 
Conservation  of  energy,  2  seq. 
Constant  boiling  mixtures,  284 
Constant,     equilibrium,     variation 
with  temperature,    143, 
seq.,  171,  376  seq. 

,  variation  with  pressure, 

172 

Continuity  of  state,  75  seq. 
Copper  sulphate,  HoO,  phase  equi- 
libria, 255 

Corresponding  states,  99 
Cryohydric  point.     See  Eutectic. 
Cycles,  thermodynamical,    29  seq., 
36  seq.,  41  seq.,  62  seq. 


D 


Deacon  process,  384  seq. 

Dehydration  of  a  salt,  293 

Depolarisation,  207 

Differential,  partial,  43 

Differentials,  complete  and  incom- 
plete, 50  seq.,  54  seq. 

Dilatometer,  265 

Dimensions,  4  seq. 

Discharge  of  ions,  mechanism  of, 
205 

Displacement  Law  of  Wien,  402 
seq. 

Dissociation  pressure,  293 

Distribution  law,  ill,  238,  239 


E 


Efficiency  of  a  heat  engine,  36  seq., 

64  seq. 
Electrical  equilibrium  (in  a  cell),  1 19 

seq.,  127  seq. 
Electrochemical  equivalent,  128 


Electrode  potential  (single),  1 74  seq. , 

177 

Electroluminescence,  408,  410 
Electrolysis  of  complex  salt,   205, 

206 
Electrolytic  dissociation,  degree  of, 

167  seq. 
-,  effect  of  pressure  on,  172 

—  potential,  204 

Electromotive  force  (of  a  concentra- 
tion cell).     See  Concentration  cell . 

Emissive  power  (for  radiation),  401 

sty. 

Enantiotropy.  295 
Energy,  capillary,  33 
,  conservation  of,  2  seq. 

— ,  dimensions  of,  4  seq. 

,  electrical,  33 

,  free.     See  Free  energy. 

— •  in    a    spectrum,    distribution 

of,  406  seq.,  475 
,  internal.     See  Internal  energy. 

—  of  rotation,  465  seq. 

—  of  translation,  464  seq. 

of  vibration,  465  seq. 

Entropy,  62  seq. 

Equations  of  state.     Cf.  Chap.  III. 
Equilibrium,  in  seq.t  129  seq. 

" box,"  132  seq. 

constant.     See  Constant. 

,  principle  of  mobile,  140  seq., 

172,  240,  265,  271 
"  Equipartition,"  of  energy,  4.64  seq. 
Eutectic  point,  273,  280,  283 
Expansion,    adiabatic.      See   Adia- 

batic  expansion. 
,   isothermal.     See  Isothermal 

expansion. 

,  work  of  gaseous,  4  seq.,  1 1  seq. 

External  work,  forms  of.    See  Work . 


Ferric  chloride,  H2O,  phase  equi- 
libria, 255,  290  seq. 


INDEX    TO   SUBJECTS 


545 


First  law  of  thermodynamics,  2  seq., 

3°.  34,  49  seq.,  56  «?•»  59  «y- 
Fluorescence,  408,  409 
Four-component  systems,  255 
Freedom,  degree  of,  241  seq.,  245 

(footnote) 

(statistical),  464  seq. 

Free  energy,   16  seq.,   32  seq.,  40, 

41,  129,  320,  324,  357  seq. 
Full  radiator,  401,  404,  405 
Fusion,  free  energy  and  internal 

energy  changes  involved  in,  367 

seq. 

G. 

Gas  cells,  195,  196 

,  calculation  of  E.M.F. 

from  thermal  data,  390  seq. 
Gas  constant  (R),  59 
per   molecule  (/£),    489 

seq. 
Gas,  perfect,  27  seq.,  58  seq. 


H 


Half  element,  202  seq. 

Heat,  latent.    See  Latent  heat. 

,  molecular  (of  water  of  crystal- 
lisation), 372 

,  specific.     See  Specific  heat. 

—  theorem  of  Nernst,   149,   357 
iff. 

• • in  relation  to  the 

quantum  theory,  524  seq. 

Hydrolysis  (of  salts),  167  seq. 

,  electrometnc  determina- 
tion, 211  seq. 


Ice,  specific  heat  of,  369 

,  varieties  of,  259  seq. 

T.P.C. — II. 


Ice- water  equilibrium,  115  seq.,  256 
Induction,  photochemical,  418  seq. 
Internal  energy,  7  seq.,  40,  41,  50, 
84  seq.,  357  seq. 

work,  77,  79,  80,  101  seq. 

Inversion  point  (of  a  gas),  82,  83, 

89  seq. 
Iodine  vapour,  dissociation  of,  385 

seq. 
Irreversible  process,  10,  2$seq.,  31, 

463 
"  Isochore  of  van  't  Hoff,"  143  seq., 

146  seq.,  170  seq. 

"Isotherm  of  van  't  Hoff,"  137  seq. 
Isothermal  expansion,  29  seq. 


Latent  heat,  8,  13,  44  seq.,  58  seq., 
66  seq.,  68  seq.,  75  seq.,  77,  79, 
80,  95,  97  seq.,  105  seq.,  166,  167 

Light  radiations,  source  of.  Re- 
source. 

Lime-carbon  dioxide,  phase  equi- 
libria, 255 

Liquid  crystals,  285 

mixtures.     See  Mixtures. 

potential  difference,  182  seq., 

189,  190,  196  seq. 

,  elimination  of,  189 

Luminescence,  398,  401,  407  seq. 


M. 

Mass  action,  deduction  of  the  law 
of,  133  seq.,  170 

,  constant,  variation  with 

temperature,  143  seq.,  171 

Maximum  work.  See  Work,  maxi- 
mum. 

Melting  point,  effect  of  pressure  on, 

47 

of  stable  and   unstable 

modifications,  125  seq.,  297  seq. 

2    N 


546 


INDEX    TO   SUBJECTS 


Membrane  equilibria,  309  seq. 

" hydrolysis,"  315  seq. 

" potential,"  318  seq. 

,  semipermeable.  See  Semiper- 

meable  membrane. 

Mixtures,  liquid,  235  seq.,  283  seq. 

Mobile  equilibrium,  principle  of. 
See  Equilibrium* 

Molecular  weight  of  dissolved  sub- 
stances, 164^.,  171 

Molecules  in  one  gram-molecule, 
number  of,  489 

Monotropy,  295 


O 


256 


One-component  systems,    255. 
seq. 

Osmotic  equilibrium,  117 

law,  deduction  of,  150  seq. 

pressure    and    electromotive 

force,  1 74  seq. 

. .  and  lowering  of  freezing 

point,  162  seq.,  1 66 

. and  lowering  of  solu- 
bility (of  the  solvent 
in  a  second  liquid), 

159  seq.,  166 

and  lowering  of  vapour 

pressure  of  a  concen- 
trated solution,  221 
seq.,  22$  seq. 

. and  lowering  of  vapour 

pressure  (of  a  dilute 
solution),  157  seq., 
165  seq. 

. and  rise  of  boiling  point, 

160  seq.,  167 

-  influence  of  hydrostatic 

pressure  on,  228  seq. 

of  compressible  solutions 

of  any  degree  of  con- 
centration (Porter's 
theory),  224  seq. 

of  concentrated  solutions, 

217  seq. 


Osmotic  theory  of  cells,  174  seq. 

work,  13  seq. 

Oxidation  processes,  affinity  of,  349 


seq. 


Perfect  gas,  thermodynamic  defini- 
tion of,  27  seq.,  68,  88,  89 
Periodic  law  and  characteristic  vi- 
bration frequency,  510  seq. 
Phase,  definition  of,  241  sfq. 
Phase  rule,  ill  sty.,  131,  240  seq., 

249  ssq. 

Phosphorescence,  408,  409 
Phosphorus,   phase   equilibria,  299 

seq. 

Photo-catalysis,  417,  420 
Photochemical,  "after  effect,"  423 
—  efficiency,  43 1  seq. 
-   equivalent,  law   of   the,   537 
seq. 

reactions,      electromagnetic 

theory  of,  454  seq. 

,  thermodynamical  theory 

of,  443  seq.,  459  seq. 
— ,  types  of,  415 
—  reversible  reactions,  423.  425 
seq.,  443  seq. 

sensitisers.     See  Sensitisers. 

Photo-electric  effect,  493  seq. 

,  selective.     See  Selective. 

Photokinetics,  410  seq.,  412  seq. 
Photoluminescence,  408,  409 
Photo,  "  stationary  state,"  413,  448, 

45°»  45 !»  452>  456»  460 

Photovoltaic  cells,  433  seq. 

Polarisation,  207 

Porous  plug  experiment,  82  seq.,  89 
seq. 

Potential  at  an  electrode.  See  Elec- 
trode. 

Principle  of  "maximum  work" 
(Berthelot),  322 

Pseudo-binary  substance,  297  seq. 


INDEX  TO   SUBJECTS 


547 


Quantum  constant  "  A,"  489 
,  size  of  a,  491  seq. 

—  theory,  395,  475  seq.,  540  seq. 
and  Nernst's  heat  theorem, 

524  seq.. 


Radiant  energy,  quantum  theory  of. 

See  Quantum. 
Radiation  due  to  temperature.     Sec 

Temperature  radiation. 

formula  of  Planck,  483  seq. 

Radiators  (vibrators),  optical,  397 

seq.,  478  seq. 

"  Reciprocal  salt  pair,"  255 
Reduced  equation  of  state,  90  seq. 
Reduction  processes,  affinity  of,  349 

seq. 

Regelation  of  ice,  47,  265 
Reversible  cycle.     See  Cycles. 
process,  9  seq.,  \%seq.,  23  seq., 


Saturated  steam,  107  seq. 

Second  law  of  thermodynamics,  34 

seq.,  38  j^.,  60  j^.,  65 
"  Selective    photo-electric    effect," 

523 

Semi-permeable  membrane,  13  seq. 
Sensitisers,  (photo)  chemical,  416, 

417 

,  optical,  420,  421,  422 

Silver  salts,  photo-reactivity  of,  426 
Sodium  sulphate,  H2O,  phase  equi- 
libria, 255,  286  seq. 
Solid  solutions,  275  seq. 
Solubility  and  osmotic  pressure,  159 

determination    by   means  of 

E.M.F.,  208  seq. 


Solubility  of  stable  and  unstable 
allotropic  modifications,  124 

Solution  pressure  (of  a  metal),  174, 
I'll  seq. 

Solutions,  theory  of  dilute,  Chap. 
VI. 

,  concentrated,  Chap. 

VIII. 

Source  of  light  radiations.  Sec 
Radiators. 

Specific  heat,  25  seq.,  57  seq.,  66  seq., 
70  seq.,  75  seq.,  78, 
8 1  seq.,  107  seq.,  361 
seq.,  468  seq. 

of  solids  at  low  tem- 
peratures, 503  seq. 

of  solids,  quantum  theory 

of,  497  seq. 

Stable  and  unstable  forms  (allo- 
tropic modification),  122  seq. 

Standard  electrode.  See  Half  ele- 
ment. 

"  Stationary  state "  (photo- 
chemical), 413 

Steel,  255 

Sublimation,  270 

Sulphur,  phase  equilibria,  255,  266 
seq.,  295  seq. 

Sulphuric  acid-water  system,  I  ibstq. 


Temperature  radiation,  398  seq. 

,  laws  of,  402-406 

Tensimeter,  266 

Thermodynamic    criteria    of   equi- 
librium, Chap  IV. 

potential,  113  seq.,  250  seq. 

Thermodynaruical      cycles.         See 

Cycles. 

Thermodynamics,  first  law  of.     See 
First  law. 

— ,  second  law  of.     See  Second 

law. 
Thermoluminescence,  408,  409 


548 


INDEX   TO  SUBJECTS 


Three-component  systems,  255 
Tin,    allotropic    modifications    of, 

127^.,  255,  271  ^.,299 
Total  energy.     See  Internal  energy. 
Transition  point  (temperature),  121 

seq.,  127,  247,  365  seq. 
Triboluminescence,  408,  409 
Triple  points  in  ice-  water  system, 

260 
Two-component  systems,  255,  272 

seq. 


U 


Ultra-violet  and  infra-red  vibration 
frequency,  relation  between,  523 
seq. 

Unary  substance,  296  seq. 


Valency  of  ions,  electrometric 
method  of  determining,  210,  211 

Vaporisation,  work  of.  See  Work 
of  vaporisation. 

Vapour,  pressure  of  saturated,  93 
seq.,  95  seq. 

• over  stable  and  un- 
stable forms,  122  seq. 

Vapour  pressure,  lowering  of  (of 
a  solution),  157  seq. 


Vapour  pressure  with  hydrostatic 
pressure,  variation  of,  230  seq. 

Variables  of  a  system,  245  seq. ,  249 
seq. 

Vibration  frequency,  characteristic 
of  a  solid,  methods  of  determin- 
ing, 506  seq. 

Virtual  work,  principle  of,  1 14  seq. 

Voltaic  cells  consisting  of  liquid  or 
solid  substances,  373  seq. 


W 

Water,  phase  equilibria,   255,   256 
seq. 

vapour,   thermal  dissociation 

of,  382  seq. 

,  photochemical  dissocia- 
tion of,  427  seq. 

Work,  definition  of,  3,  50 

expression,  van  't  Hoff  s.     See 

Isotherm  of  van  't  Hoff. 

,  external,  forms  of,  32  seq.,  119 

,  maximum,  8  seq.,  13  seq.,   18 

seq.,  36  seq.,  38  seq. 

of  expansion,  4  seq.,  II   seq., 

38  seq.,  41  seq.,  50 

of  vaporisation,    12,    13,    69 

seq. 

,  osmotic.     See  Osmotic  work. 

,  virtual.     See  Virtual  work. 


INDEX    TO    NAMES    OF    AUTHORS 


Abegg,  168,  180,  182 
Allmand,  174 
Amagat,  85,  86 
Aschkinass,  506 
Auerbach,  203 
Avogadro,  154,  491 


B 


Bakker,  IOO  seq. 

Bancroft,  196,  240,  443,  444 

Barker,  97,  308,  367 

Baur,  E.,  433,  439^- 

Beer,  411 

Berkeley,    Earl  of,  216   seq.,   228, 

235 

Berthelot,  M.,  322  seq.,  388  seq. 
Berzelius,  320 
Biltz,  W.,  510^.,  521 
Bjerrum,  199  seq.,  466 
Bodenstein,    328,    385    seq.,    419, 

537  seq. 

Boudouard,  336 
Boyle,  23,  84  seq.,  154 
Braun,  140  seq.,  143  seq.,  157,  172, 

240,  258,  265 
Bredig,  215,  221,  334 
Bridgman,  P.,  262  seq. 
Brill,  388,  389 
Bronsted,  365  seq. 
Brown,  II.  T.,  431  seq. 
Bruner,  421,  423 


Bunsen,  155  seq.,  417  seq. 

Burgess,  419 

Burke  (Miss)  K.  A.,  331,  368 

Burton,  C.  V.,  218 

Byk,  443  seq.,  454  seq. 


Carnot,  36,  37,  41,  42,  62  seq.,  65 

Carrara,  195 

Carson,  296 

Coblentz,  403,  485 

Coehn,  427  seq. 

Cohen,  E.,  190,  193,  272,  295,  302 

Chad  wick,  425 

Chapman,  301,  419,  425 

Chwolson,  69,  140 

Ciamician,  416 

Clapeyron,  44,  45,  46,  48,  68  seq., 

95»  9§>  376 
Clark,  373  seq. 
Clausius,  35,  65,  79 
Gumming,  180,  182,  201  seq. 


D 


Daniell,  41,  119-121,  339  seq.,  354, 

358 

Deacon,  384,  385 
Debye,  530  seq. 
Denham,  H.  G.,  211  seq.,  215 
Desch,  241,  282 
Dewar,  100 


549 


550  INDEX    TO   NAMES    OF  AUTHORS 

II 


Dieterici,    78,    So,    92,     105   seq., 

221 

Dixon,  425 
Dolezalek,  236 


|    Haber,  39,  208,  381,  523 
j    Harris,  313  seq. 


Donnan,  151,  303,  309  seq.,  317         !    Hartley,  216  seq. 


Draper,  417,  430 
Drude;  399 
Dulong,  473,  501 
Dunningham,  255 
Dunstan,  266 


Easley,  237 

Einstein,   395,    475   seq.,  497   seq., 
508,    514    sty.,    524    seq.,    531 

537  «?• 
Escombe,  431 
Eucken,  504 


Falcke,  336 
Fanjung,  172 
Findlay,  241,  243,  266 
Fischer,  U.,  525  seq. 
Fittig,  371 
Forbes,  430 
Franck,  496 
Fraser,  154,  216 
Frauenhofer,  401 
Freundlich,  H.,  305 


Gay-Lussac,  27,  29,  39,  154 
Gibbs,    Willard,    18,  40,  41,   131, 
145,    240,    245,    303,    304   seq., 

354 

Goodwin,  209 
Grotthus,  411,  417 
Guldberg,  99,  135,  320,  325 
Guye,  1 02 


Helmholtz,    18,  40,  41,   129,   145, 

191  seq.,  320,  322,  354 
j    Henderson,  P.,  197  seq. ,  200  s<:q. 
Henry,  150  seq.,  154  seq. 
Hertz,  398,  496 
Hittorf,  1-88 
Hollnagel,  506 
Horstmann,  322 


J 


Jahn,  168' 

Jeans.  464,  477,  481,  490  seq. 

Johnson,  K.  R.,  200 

Jolibois,  299,  302 

Joule,    6,    27,  28,  29,  39,  58,  68, 

82  seq.,  88,  358 
Juttner,  525 


Kelvin,  Lord,  29,  82  seq.,  88,  358 
Kirchhoff,  25,  26,  361  seq.,  401 
Knox,  146-8,  168-169 
Knuppfer,  344 
Koenigsberger,  522 
Koref,  366,  503 
Kriiger,  395,  540 
Kuenen,  78,  245,  285 
Kurlbaum,  405,  485,  487,  506 


Ladenbnrg,  494  seq. 
Langley,  405 
Le  Blanc,  174,  195,  349 
Le  Chatelier,  140.^.,  143  ^?->  J57i 
172,  240,  258,  265,  382 


INDEX   TO   NAMES    OF  AUTHORS 


55' 


Leeuw,  de,  272 

Lehfeldt,  174,  178 

Lehmann,  286 

Lenard,  409,  494 

Lewis,  G.  N.,  204,  393 

Lewis,  W.  C.  M.,  105,  195,  307 

Lindemann,    395,    400,    482,    503, 

508^.,  515  seq.,  523,  534 
Longuinine,  97 

Lummer,  403, 405,  407, 480,  485  seq. 
Luther,    179,   410.  seq.,  414.    4*6, 

423,    426,    430,    444,   449  seq., 

458  seq. 


M 


Magnus,  A.,  530 

Maxwell,  J.  Clerk,  94,  95,  467,  477 

McEwan,  221 

Mees,  412 

Mellor,  418 

Menzies,  165 

Merritt,  426 

Meyer,  G.,  194 

Meyerhoffer,  240 

Millikan,  491 

Milner,  105 

Mitscherlisch,  320 

Moore,  B.,  431 

Morse,  154,  216 

Morton,  253 

Mullet,  J.  A.,  253,  254 


N 


Nacken,  290 

Nernst,  2,  96  etc.,  120,  149, 
175  seq.,  182  seq.,  189  seq.,  195, 
238  seq.,  328,  336,  345,  357  seq., 
361  seq.,  376  seq.,  382  seq., 
388  seq.,  395,  427,  470  ^.,502, 
503^.,  5i5^-»  524^.,  534 

Nichols,  426,  506 

Noyes,  221 


Ogg,  210 
Olie,  302 
Orr,  31 

Ostwald,  155,  168 


Partington,  237 

Paschen,  405,  485  seq. 

Pawlow,  303 

Perrin,  399,  491 

Peters,  352  seq.,  436 

Petit,  473,  501 

Pfeffer,  164 

Philip,  J.  C.,  281 

Pier,  468 

Planck,  172,  197  seq.t2ooseq.,  375, 

395,  406,  463,  475  seq.,  483  seq., 

493  seq->  497  seq.,  524,  540  seq. 
Plotnikow,  419,  423 
Pollitzer,  357 
Porter,  A.  W.,  89,  91,  92,  114,  218, 

224  seq. 
Pousot,  325 
Preston,  402,  407 
Pringsheim,     398,    403,    405,    407, 

480,  485  seq. 


R 


Ramsay,  78  seq.,  513 
Ramsbottom,  425 
Raoult,  159,  164,  1 66 
Rayleigh,  Lord,  151,  322,  406,  407, 

479.  491 

Regnault,  78,  366 
Reicher,  266 
Reinganum,  78 
Reinitzer,  285 
Rice,  J.,  475 
Richardson,  O.  W.,  540 


552  INDEX   TO   NAMES    OF  AUTHORS 

Roozeboom,  240,  290 
Rosanoff,  237 
Roscoe,  284,  417  seq. 
Rose-Innes,  84 
Roshdestwensky,  195 
Roth,  168 
Rowlands,  6 

Rubens,  407,  485,  491,  506 
Rum  ford,  6 


Sackur,    155    seq.,    191,   219    seq. 

221  seq.,  327,  525 
Sander,  155 
Schafer,  440 
Scheel,  371 
Schenk,  336 
Schiller,  H.,  433,  438 
Schottky,  330,  370  seq. 
Schumann,  492 
Semiller,  336 
Sheppard,  395,  412 
Silber,  416 
Smith,  A.,  295  seq. 
Smits,  A.,  272,  289,  295  seq. 
Starck,  385 
Stark,  487,  542 
Stefan,  402,  405.  483,  488 
Stephenson,  J.,  218 
Stern,  155  seq.,  218,  236 
Stewart,  Balfour,  401 
Stokes,  408 


Tammann,  259,  260,  365 
Thomsen,  Julius,  322,  331,  368  seq. 
Thomson,  Sir  J.  J.,  230,  232,  478, 

494  seq. 
Thomson,  Sir  William.     See  LORD 

KELVIN. 

Thomson,  James,  93 
Titlestad,  433  seq. 


Traube,  J.,  102 
Trautz,  395,  409 
Trouton,  97  seq. 
Tswett,  431 
Tyndall,  402 


U 


Urbain,  409 


V 


van  't  Hoff,  132  seq.,  137  seq., 
143  seq.,  146  seq.,  150^.,  166, 
170  seq.,  216,  223,  228,  320  seq., 
344 

van  der  Waals,  77  seq.,  81,  90  seq., 
94  seq.,  98  seq.,  218 

W 

Waage,  135,  320,  325 

Waentig,  426 

Wartenberg,  voa,    328,    336,   393, 

427,  5°7>  527 
Washburn,  236 
Weigert,  414,  417,  419,  422,  432, 

444,  449  seq.,  458  seq. 
Wiedemann,  78,  399 
Wien,  402  seq.,  406,  407,  483 
Wigand,  366 
Wilhelmy,  320 
Wilsmore,  209 
Winther,  42 1 
Wuite,  289 


Young,  S.,  78  seq.,  105,  285 


Zawadski,  208 
Zeuner,  69 


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